(a) For the Bayesian model given by ( | )~ ( , ) Y Bin n and the prior

~ (,) Beta , find the posterior predictive density of a future data

value x, whose distribution is defined by ( | , )~ ( , ) x y Bin m .

(b) A bent coin is tossed 20 times and 6 heads come up. Assuming a flat

prior on the probability of heads on a single toss, what is the probability

that exactly one head will come up on the next two tosses of the same

coin? Answer this using results in (a).Chapter 3: Bayesian Basics Part 3

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(c) A bent coin is tossed 20 times and 6 heads come up. Assume a

Beta(20.3,20.3) prior on the probability of heads.

Find the expected number of times you will have to toss the same coin

again repeatedly until the next head comes up.

(d) A bent coin is tossed 20 times and 6 heads come up. Assume a

Beta(20.3,20.3) prior on the probability of heads.

Now consider tossing the coin repeatedly until the next head, writing

down the number of tosses, and then doing all of this again repeatedly,

again and again.

The result will be a sequence of natural numbers (for example

3, 1, 1, 4, 2, 2, 1, 5, 1, ....), where each number represents a number of

tails in a row within the sequence, plus one.

Next define to be the average of a very long sequence like this (e.g.

one of length 1,000,000). Find the posterior predictive density and mean

of (approximately)

6. Let X1, X2, X3 be a random sample from the uniform distribution on the

interval (0, 1). What is the probability that the sample median is less than

0.4?

7. Let X1, X2, X3, X4, X5 be a random sample from the uniform distribution

on the interval (0, ), where is unknown, and let Xmax denote the largest

observation. For what value of the constant k, the expected value of the

random variable kXmax is equal to ?

8. A random sample of size 16 is to be taken from a normal population having

mean 100 and variance 4. What is the 90th percentile of the distribution of

the sample mean?

4. Suppose that P(An B) = 0.2, P(A) = 0.6 and P(B) = 0.5. Find P(AcUBC), P(Acn B"), P(An B) and P(AUB) 5. A sample space consists of five simple events, E1, E2, E3, E4, Es. . If P(E1) = P(E2) = 0.15, P(E3) = 0.4, P(E4) = 2P(Es) find the probabilities of E4 and Es. . If P(E1) = 3P(E2) = 0.3, find the probabilities of the remaining simple events if you know that the remaining simple events are equally proba- ble.Part A. For two independent events A and B, suppose that P(A) = 0.2 and P(B) = 0.4. To find P(A /7 B), what probability rule/law, should you use? O Axioms of probability O Definition of the independent events O The law of total probability O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The addition rule Find P(A /) B). To find P(A U B), what probability rule/law, should you use? O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The law of total probability O The addition rule O Definition of the independent events O Axioms of probability Find P(A U B).2.4 Counting Rules Useful in Probability 31 2.26 Consider the information given in Exercise 2.24. In a poll of randomly selected veterinarian services workers, it is desired to get responses from approximately 50 Blacks or African Americans. How many veterinarian services workers should be polled? 2.27 For events A and B, using the axioms of probability, prove that P(AB) P(A) + P(B) - 1. 2.29 For events A and B, using the axioms of probability, prove that P(AB) = P(A) - P(AB). 2.30 Using the axioms of probability, show that for events A, B, and C, P(AUBUC) = P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC). 2.31 Suppose that an experiment is performed n times. For any event A of the experiment, let n(A) denote the number of times that event A occurs. The relative frequency definition of probability would propose that P(A) = n(A). Prove that this definition satisfies the three axioms of probability. 2.32 For the n events, E1, Ez, . .., E,,, use induction to prove that P ( U E , ) = [ P (E; ) - [ P(E; En ) + .. 1= 1 in