Introduction to probability joseph k. blitzstein pdf download free

In this case, instead of pebbles, we can visualize mud spread out over a region, where the total mass of the mud is 1. Any function P mapping events to numbers in the interval [0, 1] that satisfies the two axioms is considered a valid probability function. Regardless of how we choose to interpret probability, we can use the two axioms to derive other properties of probability, and these results will hold for any valid probability function.

Probability has the following properties, for any events A and B. This is illustrated in the Venn diagram below. The intuition for this result can be seen using a Venn diagram like the one below. This is a useful intuition, but not a proof. For intuition, consider a triple Venn diagram like the one below. Finally, the region in the center has been added three times and subtracted three times, so in order to count it exactly once, we must add it back again.

This ensures that each region of the diagram has been counted once and exactly once. Now we can write inclusion-exclusion for n events. For any events A1 ,. Mixed practice A patient is being given a blood test for the disease conditionitis.

Let p be the prior probability that the patient has conditionitis. The blood sample is sent to one of two labs for analysis, lab A or lab B. Conditional probability For lab A, the probability of someone testing positive given that they do have the disease is a1 , and the probability of someone testing negative given that they do not have the disease is a2. The corresponding probabilities for lab B are b1 and b2. Fred decides to take a series of n tests, to diagnose whether he has a certain disease any individual test is not perfectly reliable, so he hopes to reduce his uncertainty by taking multiple tests.

Let Tj be the event that he tests positive on the jth test. Find the posterior probability that Fred has the disease, given that he tests positive on all n of the n tests. However, some people have a certain gene that makes them always test positive. Let G be the event that Fred has the gene.

If Fred does not have the gene, then the test results are conditionally independent given his disease status. Find the posterior probability that Fred has the disease, given that he tests positive on all n of the tests.

A certain hereditary disease can be passed from a mother to her children. A week later, the younger child is also found not to have the disease. Given this information, find the probability that the mother has the disease. Three fair coins are tossed at the same time. An urn contains red, green, and blue balls. Find the probability that the first time a green ball is drawn is before the first time a blue ball is drawn. Hint: Explain how this relates to finding the probability that a draw is green, given that it is either green or blue.

Is the answer the same or different than the answer in a? Hint: Imagine the balls all lined up, in the order in which they will be drawn. Note that where the red balls are standing in this line is irrelevant. Independent trials are performed, and the outcome of each trial is classified as being exactly one of type 1, type 2,. The robbers offer you the option of drawing first, last, or at any time in between. When would you take your turn? The draws are made without replacement, and for a are uniformly random.

Give a clear, concise, and compelling explanation. At each stage, draws are made with probability proportional to weight, i. A fair die is rolled repeatedly, until the running total is at least at which point the rolling stops. Find the most likely value of the final running total i. Hint: Consider the possibilities for what the running total is just before the last roll. Homer has a box of donuts, which currently contains exactly c chocolate, g glazed, and j jelly donuts.

Homer eats donuts one after another, each time choosing uniformly at random from the remaining donuts. Hint: Consider the last donut remaining, and the last donut that is either glazed or jelly. Let D be the event that a person develops a certain disease, and C be the event that the person was exposed to a certain substance e.

We are interested in whether exposure to the substance is related to developing the disease and if so, how they are related. P D C c The relative risk is especially easy to interpret, e. A researcher wants to estimate the percentage of people in some population who have used illegal drugs, by conducting a survey. The slip is then returned to the hat. The researcher does not know which type of slip the participant had. Find d, in terms of y and p. At the beginning of the play Rosencrantz and Guildenstern Are Dead by Tom Stoppard [25], Guildenstern is spinning coins and Rosencrantz is betting on the outcome for each.

The coins have been landing Heads over and over again, prompting the following remark: Guildenstern: A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability.

The coin spins have resulted in Heads 92 times in a row. Upon seeing the events described above, they have the following conversation: Introduction to Probability Fred : That outcome would be incredibly unlikely with fair coins. They must be using trick coins maybe with double-headed coins , or the experiment must have been rigged somehow maybe with magnets. The reason the outcome seems extremely unlikely is that the number of possible outcomes grows exponentially as the number of spins grows, so any outcome would seem extremely unlikely.

Discuss these comments, to help Fred and his friend resolve their debate. Let p be the prior probability that the coins are fair. Find the posterior probability that the coins are fair, given that they landed Heads in 92 out of 92 trials.

For which values of p is it less than 0. There are n types of toys, which you are collecting one by one. Each time you buy a toy, it is randomly determined which type it has, with equal probabilities.

This problem is in the setting of the coupon collector problem, a famous problem which we study in Example 4. For example, a company may want to see how users will respond to a new feature on their website compared with how users respond to the current version of the website or compare two different advertisements. As the name suggests, two different treatments, Treatment A and Treatment B, are being studied. Users arrive one by one, and upon arrival are randomly assigned to one of the two treatments.

The probability that the nth user receives Treatment A is allowed to depend on the outcomes for the previous users. This set-up is known as a two-armed bandit. Many algorithms for how to randomize the treatment assignments have been studied.

Here is an especially simple but fickle algorithm, called a stay-with-a-winner procedure: i Randomly assign the first user to Treatment A or Treatment B, with equal probabilities. Let a be the probability of success for Treatment A, and b be the probability of success for Treatment B. Let pn be the probability of success on the nth trial and an be the probability that Treatment A is assigned on the nth trial using the above algorithm. In humans and many other organisms , genes come in pairs.

A certain gene comes in two types alleles : type a and type A. The genotype of a person for that gene is the types of the two genes in the pair: AA, Aa, or aa aA is equivalent to Aa.

A standard deck of cards will be shuffled and then the cards will be turned over one at a time until the first ace is revealed. Let B be the event that the next card in the deck will also be an ace. Explain your intuition. Give an intuitive discussion rather than a mathematical calculation; the goal here is to describe your intuition explicitly.

Find P B Cj in terms of j, fully simplified. The sum can be left unsimplified, but it should be something that could easily be computed in software such as R that can calculate sums. Hint: If you were deciding whether to bet on the next card after the first ace being an ace or to bet on the last card in the deck being an ace, would you have a preference?

Random variables are essential throughout the rest of this book, and throughout statistics, so it is crucial to think through what they mean, both intuitively and mathematically. In this problem, we may be very interested in how much wealth each gambler has at any particular time. So we could make up notation like letting Ajk be the event that gambler A has exactly j dollars after k rounds, and similarly defining an event Bjk for gambler B, for all j and k.

This is already too complicated. Let Xk be the wealth of gambler A after k rounds. The notion of a random variable will allow us to do exactly this! It needs to be introduced carefully though, to make it both conceptually and technically correct.

To make the notion of random variable precise, we define it as a function mapping the sample space to the real line. See the math appendix for review of some concepts about functions. The r. X depicted here is defined on a sample space with 6 elements, and has possible values 0, 1, and 4. The randomness comes from choosing a random pebble according to the probability function P for the sample space.

Definition 3. Given an experiment with sample space S, a random variable r. It is common, but not required, to denote random variables by capital letters. Thus, a random variable X assigns a numerical value X s to each possible outcome s of the experiment.

The randomness comes from the fact that we have a random experiment with probabilities described by the probability function P ; the mapping itself is deterministic, as illustrated in Figure 3. The same r. This definition is abstract but fundamental; one of the most important skills to develop when studying probability and statistics is the ability to go back and forth between abstract ideas and concrete examples. Relatedly, it is important to work on recognizing the essential pattern or structure of a problem and how it connects Random variables and their distributions to problems you have studied previously.

We will often discuss stories that involve tossing coins or drawing balls from urns because they are simple, convenient scenarios to work with, but many other problems are isomorphic: they have the same essential structure, but in a different guise.

The structure of the problem is that we have a sequence of trials where there are two possible outcomes for each trial. Example 3. Consider an experiment where we toss a fair coin twice. Here are some random variables on this space for practice, you can think up some of your own. Each r. This is a random variable with possible values 0, 1, and 2.

This r. For most r. Fortunately, it is usually unnecessary to do so, since as we saw in this example there are other ways to define an r. A random variable simply labels each pebble with a number. Figure 3. After we perform the experiment and the outcome s has been realized, the random variable crystallizes into the numerical value X s. Random variables provide numerical summaries of the experiment in question.

For example, the experiment may be to collect a random sample of people in a certain city and ask them various questions, which may have numeric e. The fact that r. In this chapter and the next, our focus is on discrete r.

Continuous r. A random variable X is said to be discrete if there is a finite list of values a1 , a2 ,. If X is a discrete r. Most commonly in applications, the support of a discrete r. In contrast, a continuous r.

It is also possible to have an r. But the starting point for understanding such r. Given a random variable, we would like to be able to describe its behavior using the language of probability.

For example, we might want to answer questions about the probability that the r. If M is the number of major earthquakes in California in the next five years, what is the probability that M equals 0? The distribution of a random variable provides the answers to these questions; it specifies the probabilities of all events associated with the r. We will see that there are several equivalent ways to express the distribution of an r.

For a discrete r. The probability mass function PMF of a discrete r. Note that this is positive if x is in the support of X, and 0 otherwise. Going back to Example 3.

Here are the r. Vertical bars are drawn to make it easier to compare the heights of different points. We roll two fair 6-sided dice. For example, 7 of the 36 outcomes s are shown in the table below, along with the corresponding values of X, Y, and T. After the experiment is performed, we observe values for X and Y , and then the observed value of T is the sum of those values. Similarly, Y is also Discrete Uniform on 1, 2,. Note that Y has the same distribution as X but is not the same random variable as X.

Two more r. If the top value is equally likely to be any of the numbers 1, 2,. Suppose we choose a household in the United States at random. Let X be the number of children in the chosen household.

Since X can only take on integer values, it is a discrete r. The probability that X takes on the value x is proportional to the number of households in the United States with x children.

Using data from the General Social Survey [23], we can approximate the proportion of households with 0 children, 1 child, 2 children, etc. Theorem 3. Let X be a discrete r.

The first criterion is true since probability is nonnegative. We claimed earlier that the PMF is one way of expressing the distribution of a discrete r. This is because once we know the PMF of X, we can calculate the probability that X will fall into a given subset of the real numbers by summing over the appropriate values of x, as the next example shows. Returning to Example 3. We have already calculated the PMF of T. There are only three values in the interval [1, 4] that T can take on, namely, 2, 3, and 4.

Knowing the PMF of a discrete r. We will introduce these named distributions throughout the book, starting with a very simple but useful case: an r.

The most famous distributions in statistics all have stories which explain why they are so often used as models for data, or as the building blocks for more complicated distributions. Thinking about the named distributions first and foremost in terms of their stories has many benefits.

It facilitates pattern recognition, allowing us to see when two problems are essentially identical in structure; it often leads to cleaner solutions that avoid PMF calculations altogether; and it helps us understand how the named distributions are connected to one another.

Here it is clear that Bern p is the same distribution as Bin 1, p : the Bernoulli is a special case of the Binomial. To save writing, it is often left implicit that a PMF is zero wherever it is not specified to be nonzero, but in any case it is important to understand what the support of a random variable is, and good practice to check that PMFs are valid.

If two discrete r. So we sometimes refer to the support of a discrete distribution; this is the support of any r. An experiment consisting of n independent Bernoulli trials produces a sequence of successes and failures. There are nk such sequences, since we just need to select where the successes are. This is a valid PMF because it is nonnegative and it sums to 1 by the binomial theorem.

For a fixed number of trials n, X tends to be larger when the success probability is high and lower when the success probability is low, as we would expect from the story of the Binomial distribution. Also recall that in any PMF plot, the sum of the heights of the vertical bars must be 1. In the lower left, we plot the Bin , 0.

Random variables and their distributions Theorem 3. Using the story of the Binomial, interpret X as the number of successes in n independent Bernoulli trials. By Theorem 3. Consistent with Theorem 3. If we instead sample without replacement, as illustrated in Figure 3. Story 3. Consider an urn with w white balls and b black balls. Let X be the number of white balls in n the sample. As with the Binomial distribution, we can obtain the PMF of the Hypergeometric distribution from the story.

The essential structure of the Hypergeometric story is that items in a population are classified using two sets of tags: in the urn story, each ball is either white or black this is the first set of tags , and each ball is either sampled or not sampled this is the second set of tags.

Furthermore, at least one of these sets of tags is assigned completely at random in the urn story, the balls are sampled randomly, with all sets of the correct size equally likely. A forest has N elk. Today, m of the elk are captured, tagged, and released into the wild.

At a later date, n elk are recaptured at random. Assume that the recaptured elk are equally likely to be any set of n of the elk, e. Instead of sampling n balls from the urn, we recapture n elk from the forest.

In a five-card hand drawn at random from a well-shuffled standard deck, the number of aces in the hand has the HGeom 4, 48, 5 distribution, which can be seen by thinking of the aces as white balls and the nonaces as black balls. In each example, the r. Story First set of tags urn elk cards white tagged ace black untagged not ace Second set of tags sampled not sampled recaptured not recaptured in hand not in hand The next theorem describes a symmetry between two Hypergeometric distributions with different parameters; the proof follows from swapping the two sets of tags in the Hypergeometric story.

Using the story of the Hypergeometric, imagine an urn with w white balls, b black balls, and a sample of size n made without replacement. Both X and Y count the number of white sampled balls, so they have the same distribution. We prefer the story proof because it is less tedious and more memorable.

The Binomial and Hypergeometric distributions are often confused. Both are discrete distributions taking on integer values between 0 and n for some n, and both can be interpreted as the number of successes in n Bernoulli trials for the Hypergeometric, each tagged elk in the recaptured sample can be considered a success and each untagged elk a failure.

However, a crucial part of the Binomial story is that the Bernoulli trials involved are independent. The Bernoulli trials in the Hypergeometric story are dependent, since the sampling is done without replacement: knowing that one elk in our sample is tagged decreases the probability that the second elk will also be tagged.

Let C be a finite, nonempty set of numbers. Choose one of these numbers uniformly at random i. Call the chosen number X. As with questions based on the naive definition of probability, questions based on a Discrete Uniform distribution reduce to counting problems.

C Random variables and their distributions Example 3. There are slips of paper in a hat, each of which has one of the numbers 1, 2,. Five of the slips are drawn, one at a time. First consider random sampling with replacement with equal probabilities.

Now consider random sampling without replacement with all sets of five slips equally likely to be chosen. Solution: a By the story of the Binomial, the distribution is Bin 5, 0. This solution just uses new notation for concepts from Chapter 1. It is useful to have this new notation since it is compact and flexible. This follows from the naive definition in this case, but a more general way to think about such statements is through independence of r. This is the unconditional distribution of Yj : we are working from a vantage point before drawing any of the slips.

For further insight into why each of Y1 ,. This formulation does not change the problem in any important way, and it helps avoid getting distracted by irrelevant chronological details. Label the five people 1, 2,. It would be bizarre if the answer to c were greater than or equal to the answer to f , since sampling without replacement makes it easier to find the number For the same reason, when searching for a lost possession it makes more sense to sample locations without replacement than with replacement.

Unlike the PMF, which only discrete r. The cumulative distribution function CDF of an r. When there is no risk of ambiguity, we sometimes drop the subscript and just write F or some other letter for a CDF. The next example demonstrates that for discrete r. For example, in Figure 3. Any CDF F has the following properties. Wherever there is a jump, the CDF is continuous from the right.

X whose possible values are 0, 1, 2,. As an example of how to visualize the criteria, consider Figure 3. To recap, we have now seen three equivalent ways of expressing the distribution of a random variable. Generally the PMF is easier to work with for discrete r. Random variables and their distributions A third way to describe a distribution is with a story that explains in a precise way how the distribution can arise.

We used the stories of the Binomial and Hypergeometric distributions to derive the corresponding PMFs. Thus the story and the PMF also contain the same information, though we can often achieve more intuitive proofs with the story than with PMF calculations. X is defined on a sample space with 6 elements, and has possible values 0, 1, and 4.

The next example illustrates this method. A particle moves n steps on a number line. The particle starts at 0, and at each step it moves 1 unit to the right or to the left, with equal probabilities. Assume all steps are independent. Find the PMF of Y. Solution: Consider each step to be a Bernoulli trial, where right is considered a success and left is considered a failure. Assume that n is even. Find the PMF of D. Given an experiment with sample space S, if X and Y are r.

Note that we are assuming that X and Y are defined on the same sample space S. Usually we assume that S is chosen to be rich enough to encompass whatever r. One way to understand the mapping from S to R represented by the r. For a less familiar example like max X, Y , students often are unsure how to interpret it as an r.

Let X be the number on the first die and Y the number on the second die. The following table gives the values of X, Y , and max X, Y under 7 of the 36 outcomes in the sample space, analogously to the table in Example 3.

Many common mistakes in probability can be traced to confusing two of the following fundamental objects with each other: distributions, random variables, events, and numbers. Such mistakes are examples of category errors. To help avoid being categorically wrong, always think about what category an answer should have. An especially common category error is to confuse a random variable with its distribution.

We call this error sympathetic magic; this term comes from anthropology, where it is used for the belief that one can influence an object by manipulating a representation of that object.

The following saying sheds light on the distinction between a random variable and its distribution: The word is not the thing; the map is not the territory. Just as different houses can share the same blueprint, different r. It does not make sense to multiply a PMF by 2, since the probabilities would no longer sum to 1. As we saw above, if X takes on values xj with probabilities pj , then 2X takes on values 2xj with probabilities pj. Note that X can take on odd values, but 2X is necessarily even.

Just because two r. We saw this in Example 3. As another example, consider flipping a fair coin once. Intuitively, if two r. The definition formalizes this idea. The definition for more than two r. Random variables X1 ,. For infinitely many r. Comparing this to the criteria for independence of n events, it may seem strange that the independence of X1 ,.

However, upon closer examination of the definition, we see that independence of r. If we can find even a single list of values x1 ,. If X1 ,. More generally, we have the following result for which we omit a formal proof. If X and Y are independent r. We will often work with random variables that are independent and have the same distribution. We call such r.

Random variables are independent if they provide no information about each other; they are identically distributed if they have the same PMF or equivalently, the same CDF. Whether two r. We can have r. Let X be the result of a die roll, and let Y be the result of a second, independent die roll.

Then X and Y are i. Let X be the result of a die roll, and let Y be the closing price of the Dow Jones a stock market index a month from now.

Then X and Y provide no information about each other one would fervently hope , and X and Y do not have the same distribution. Let X be the number of Heads in n independent fair coin tosses, and let Y be the number of Tails in those same n tosses. Let X be the indicator of whether the majority party retains control of the House of Representatives in the U.

Then X and Y are dependent, and X and Y do not have the same distribution. By taking a sum of i. Bernoulli r. Bern p. If we count the number of raised hands which is the same as adding up the Xi , we get the total number of successes.

We present three proofs, since each illustrates a useful technique. Representation: A much simpler proof is to represent both X and Y as the sum of i.

Bern p r. Random variables X and Y are conditionally independent given an r. For discrete r. For any discrete r. Independence of r. First let us show why independence does not imply conditional independence.

Random variables and their distributions Example 3. Consider the simple game called matching pennies. Each of two players, A and B, has a fair penny.

They flip their pennies independently. If the pennies match, A wins; otherwise, B wins. So X and Y are conditionally dependent given Z. Let X be the indicator of Alice calling me next Friday, Y be the indicator of Bob calling me next Friday, and Z be the indicator of exactly one of them calling me next Friday. Then X and Y are independent by assumption. Suppose that you are going to play two games of tennis against one of two identical twins. So X and Y are conditionally independent given Z.

Past games give us information which helps us infer who our opponent is, which in turn helps us predict future games! As we will see in this section, we can get from the Binomial to the Hypergeometric by conditioning, and we can get from the Hypergeometric to the Binomial by taking a limit.

A scientist wishes to study whether women or Introduction to Probability men are more likely to have a certain disease, or whether they are equally likely. A random sample of n women and m men is gathered, and each person is tested for the disease assume for this problem that the test is completely accurate. So the conditional distribution of X is Hypergeometric with parameters n, m, r.

In the elk story, we are Random variables and their distributions interested in the distribution of the number of tagged elk in the recaptured sample. By analogy, think of women as tagged elk and men as untagged elk.

This is exactly analogous to the number of tagged elk in the recaptured sample, which is distributed HGeom n, m, r. In the other direction, the Binomial is a limiting case of the Hypergeometric. As the number of balls in the urn grows very large relative to the number of balls that are drawn, sampling with replacement and sampling without replacement become essentially equivalent.

But this becomes less and less likely as N grows, and even if it is likely that there will be a few coincidences, the approximation can still be reasonable if it is very likely that the vast majority of balls in the sample are sampled only once each. The distribution of an r. The distribution of a discrete r.

A story for X describes an experiment that could give rise to a random variable with the same distribution as X. For a PMF to be valid, it must be nonnegative and sum to 1.

It is important to distinguish between a random variable and its distribution: the distribution is a blueprint for building the r. Each of these is actually a family of distributions, indexed by parameters; to fully specify one of these distributions, we need to give both the name and the parameter values.

A function of a random variable is still a random variable. Two random variables are independent if knowing the value of one r. This is unrelated to whether the two r.

In Chapter 7, we will learn how to deal with dependent random variables by considering them jointly rather than separately. We have now seen four fundamental types of objects in probability: distributions, random variables, events, and numbers. A CDF can be used as a blueprint for generating an r. Knowing the probabilities of these events determines the CDF, taking us full circle. What can happen? From a CDF F we can generate an r. From X, we can generate many other r.

There are various events describing the behavior of X. Knowing the probabilities of these events for all x gives us the CDF and in the discrete case the PMF, taking us full circle. We will also explain in general how to generate r. Typing help distributions gives a handy list of built-in distributions; many others are available through R packages that can be loaded. In general, for many named discrete distributions, three functions starting with d, p, and r will give the PMF, CDF, and random generation, respectively.

Binomial distribution The Binomial distribution is associated with the following three R functions: dbinom, pbinom, and rbinom. It takes three inputs: the first is the value of x at which to evaluate the PMF, and the second and third are the parameters n and p. For example, dbinom 3,5,0. It takes three inputs: the first is the value of x at which to evaluate the CDF, and the second and third are the parameters.

So pbinom 3,5,0. For rbinom, the first input is how many r. Thus the command rbinom 7,5,0. Bin 5, 0. For example, recall that 0:n is a quick way to list the integers from 0 to n. The command dbinom ,5,0. Hypergeometric distribution The Hypergeometric distribution also has three functions: dhyper, phyper, and rhyper.

Since the Hypergeometric distribution has three parameters, each of these functions takes four inputs. For dhyper and phyper, the first input is the value at which we wish to evaluate the PMF or CDF, and the remaining inputs are the parameters of the distribution. For rhyper, the first input is the number of r. HGeom w, b, n r. Discrete distributions with finite support We can generate r. For example, to generate 5 independent DUnif 1, 2,.

It turns out that sample is far more versatile. If we want to sample from the values x1 ,. Suppose we want realizations of i. First, we use the c function to create vectors with the support of the distribution and the corresponding probabilities. Check that this is a valid PMF using properties of logs, not with a calculator. Bob is playing a video game that has 7 levels. He starts at level 1, and has probability p1 of reaching level 2. Let X be the highest level that he reaches.

Find the PMF of X in terms of p1 ,. Random variables and their distributions 8. There are boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth as a simple expression in terms of binomial coefficients.

Hint: See the math appendix for a review of some facts about series. Then sketch their PMFs on a second set of axes. Is it possible to find two different PMFs where the first is less than or equal to the second everywhere? In other words, find discrete r. Let X, Y, Z be discrete r. Show that X and Y have the same distribution unconditionally, not just when given Z.

Let X be the number of purchases that Fred will make on the online site for a certain company in some specified time period. If the company computes the number of purchases for everyone in their database, then these data are draws from the conditional distribution of the number of purchases, given that at least one purchase is made.

This conditional distribution is called a truncated Poisson distribution. Find the CDF of an r. Find the conditional distribution of X, given that X is in B. The plane has seats, and people have booked the flight. Each person will show up for the flight with probability 0.

Find the probability that there will be enough seats for everyone who shows up for the flight. Let p be the probability that A wins an individual game, and assume that the games are independent. What is the probability that team A wins the series?

In a chess tournament, n games are being played, independently. Each game ends in a win for one player with probability 0.

Find the PMFs of the number of games ending in a draw, and of the number of players whose games end in draws. Suppose that a lottery ticket has probability p of being a winning ticket, independently of other tickets.

A gambler buys 3 tickets, hoping this will triple the chance of having at least one winning ticket. There are two coins, one with probability p1 of Heads and the other with probability p2 of Heads. One of the coins is randomly chosen with equal probabilities for the two coins. Let X be the number of times it lands Heads. You can assume that n is large for your explanation, so that the frequentist interpretation of probability can be applied.

There are n people eligible to vote in a certain election. Voting requires registration. Decisions are made independently. Each of the n people will register with probability p1. Given that a person registers, they will vote with probability p2. Given that a person votes, they will vote for Kodos who is one of the candidates with probability p3.

What is the distribution of the number of votes for Kodos give the PMF, fully simplified, or the name of the distribution, including its parameters? Random variables and their distributions Give a short proof. Let X be the number of matching cards. Is X Binomial? Is X Hypergeometric? Each of these chicks survives with probability r, independently.

What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? Give the PMFs; also give the names of the distributions and their parameters, if applicable. Show that conditional on the number of successes, all valid possibilities for the list of outcomes of the experiment are equally likely.

The company is deciding which employees to promote. What is the distribution of the number of women who get promoted? Find the distributions of the number of women who are promoted, the number of women who are not promoted, and the number of employees who are promoted. Once upon a time, a famous statistician offered tea to a lady. The lady claimed that she could tell whether milk had been added to the cup before or after the tea.

The statistician decided to run some experiments to test her claim. The lady gets to taste each and then guess which 3 were milk-first.

Assume for this part that she has no ability whatsoever to distinguish milk-first from tea-first cups of tea. Find the probability that at least 2 of her 3 guesses are correct.

She needs to say whether she thinks it was milk-first. She claims that the cup was milk-first. Find the posterior odds that the cup is milk-first, given this information. Since he knows the format of the exam in advance, Evan is trying to decide how many key terms he should study. Let X be the number of key terms appearing on the exam that he has studied. What is the distribution of X? Give the name and parameters, in terms of s. A book has n typos.

Two proofreaders, Prue and Frida, independently read the book. Let X1 be the number of typos caught by Prue, X2 be the number caught by Frida, and X be the number caught by at least one of the two proofreaders.

You can leave your answer as a sum though with some algebra it can be simplified, by writing the binomial coefficients in terms of factorials and using the binomial theorem. Hint: Does it matter whether the students declare their majors before or after the random sample is drawn? Each time A answers a question, she has probability p1 of getting it right.

Each time B plays, he has probability p2 of getting it right. Find the probability that A wins the game. There are n voters in an upcoming election in a certain country, where n is a large, even number. Let X be the number of people who vote for Candidate A.

Suppose that each voter chooses randomly whom to vote for, independently and with equal probabilities. The message is a sequence x1 , x2 ,. Since the channel is noisy, there is a chance that any bit might be corrupted, resulting in an error a 0 becomes a 1 or vice versa.

Assume that the error events are independent. If X and Y are independent, is it still possible to construct such an example? Suppose X and Y are discrete r. This means that X and Y always take on the same value. Hint: Think about simple and extreme examples. Let Y be the next day after X again represented as an integer between 1 and 7.

Do X and Y have the same distribution? What is P X For example, we can use this to simulate pairwise independent fair coin tosses using only 10 independent fair coin tosses. Then A, B, C are independent since they are based on disjoint sets of Xi. Also, at most one of these sets of Xi can be empty. In a pilot study, the new treatment is given to 20 random patients, and is effective for 15 of them. Given the results of the first study, what is the PMF for how many of the new patients the new treatment is effective on?

Letting p be the answer to a , your answer can be left in terms of p. An important question that often comes up in such settings is how many trials to perform. Many controversies have arisen in statistics over the issue of how to analyze data coming from an experiment where the number of trials can depend on the data collected so far. However, it might never happen that there are more than twice as many failures as successes; in this problem, you will find the probability of that happening.

Assume that the gamblers are allowed to borrow money, so they can and do gamble forever. Explain how this story relates to the original problem, and how the original problem can be solved if we can find pk. The solution can be written neatly in terms of the golden ratio. A copy machine is used to make n pages of copies per day.

The machine has two trays in which paper gets loaded, and each page used is taken randomly and independently from one of the trays. At the beginning of the day, the trays are refilled so that they each have m pages. In terms of pbinom, find a simple expression for the probability that both trays have enough paper on any particular day, when this probability is strictly between 0 and 1 also specify the values of m for which the probability is 0 and the values for which it is 1.

Hint: Be careful about whether inequalities are strict, since the Binomial is discrete. Hint: If you use R, you may find the following commands useful: g 0.

For example, we can say how likely it is that the r. Given a list of numbers x1 , x2 ,. The definition of expectation for a discrete r. Definition 4. The expected value also called the expectation or mean of a discrete r. X whose distinct possible values are Introduction to Probability x1 , x2 ,. In words, the expected value of X is a weighted average of the possible values that X can take on, weighted by their probabilities.

Let X be the result of rolling a fair 6-sided die, so X takes on the values 1, 2, 3, 4, 5, 6, with equal probabilities.

Intuitively, we should be able to get the average by adding up these values and dividing by 6. Note that X never equals its mean in this example.

This is similar to the fact that the average number of children per household in some country could be 1. This is illustrated in Figure 4. Here q and p denote the masses of the two pebbles. We will look at a more mathematical version of this example when we study the law of large numbers in Chapter This follows directly from the definition, but is worth recording since it is fundamental. Proposition 4.

If X and Y are discrete r. Figure 4. X, the expected value E X is a number if it exists. A common mistake is to replace an r. Notation 4. We often abbreviate E X to EX. Paying attention to the order of operations is crucial when working with expectation.

As stated above, EX 2 is the expectation of the random variable X 2 , not the square of the number EX. Unless the parentheses explicitly indicate otherwise, 1. Theorem 4. For any r.

The second equation says that we can take out constant factors from an expectation; this is both intuitively reasonable and easily verified from the definition.

What may be surprising is that it holds even if X and Y are dependent! To build intuition for this, consider the extreme case where X always equals Y. Linearity is true for all r. Before proving linearity, it is worthwhile to recall some basic facts about averages. The second edition adds many new examples,. Search this site. Do you want to read it?? Subscribe to posts. Hear Book. Samuel Mayer. By - Giovanni Rigters. Binnall Book. PDF] Comprende?

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