26 Jun stats quesions answers BH 3.2 BH 3.16 BH 3.20 3.22 3.24 3.34 ETC
Question
6. (BH 3.2)
(a) Independent Bernoulli trials are performed, with probability 1=2 of success, until there has been at least one success. Find the PMF of the number of trials performed.
(b) Independent Bernoulli trials are performed, with probability 1=2 of success, until there has been at least one success and at least one failure. Find the PMF of the number of trials performed.
7. (BH 3.16) Let X DUnif(C), and B be a nonempty subset of C. Find the conditional distribution of X, given that X is in B.
8. (BH 3.20) 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.
(a) What is the distribution of how many of the 3 tickets are winning tickets?
(b) Show that the probability that at least 1 of the 3 tickets is winning is 3p 3p2 +p3, in two di erent ways: by using inclusion-exclusion, and by taking the complement of the desired event and then using the PMF of a certain named distribution.
(c) Show that the gambler’s chances of having at least one winning ticket do not quite triple (compared with buying only one ticket), but that they do approximately triple if p is small.
9. (BH 3.22) 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). It is then ipped n 2 times. Let X be the number of times it lands Heads.
(a) Find the PMF of X.
(b) What is the distribution of X if p1 = p2?
(c) Give an intuitive explanation of why X is not Binomial for p1 6= p2 (its distribu-tion is called a mixture of two Binomials). You can assume that n is large for your explanation, so that the frequentist interpretation of probability can be applied.
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10. (BH 3.23) 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, he or she will vote with probability p2. Given that a person votes, he or she 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 simpli ed, or the name of the distribution, including its parameters)?
11. (BH 3.24) Let X be the number of Heads in 10 fair coin tosses.
(a) Find the conditional PMF of X, given that the rst two tosses both land Heads.
(b) Find the conditional PMF of X, given that at least two tosses land Heads.
12. (BH 3.34) There are n students at a certain school, of whom X Bin(n; p) are Statistics majors. A simple random sample of size m is drawn (simple random sample” means sampling without replacement, with all subsets of the given size equally likely).
(a) Find the PMF of the number of Statistics majors in the sample, using the law of total probability (don’t forget to say what the support is). You can leave your answer as a sum (though with some algebra it can be simpli ed, by writing the binomial coe cients in terms of factorials and using the binomial theorem).
(b) Give a story proof derivation of the distribution of the number of Statistics majors in the sample; simplify fully.
Hint: Does it matter whether the students declare their majors before or after the random sample is drawn?
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