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[answered] Math 180A - Homework 10 ( Due Friday, Dec 2, 4:00 PM) Readi

**Question 1, 3, 5, 6, 9 in the attached file**

1. Suppose X is has a Normal distribution with mean ? and variance ?2.(a) Define Y = ?X. Find the probability density function of Y .(b) Given an example of two random variable with Normal distribution, such that their sumdoes not have a Normal distribution.

3. A group of 10 students volunteered to work abroad. Each student chooses a country atrandom among 8 possible countries. If the choices are made independently, calculate theexpected number of countries which will host no volunteers.

5. Fix a real number a.(a) For any random variable X, use the Markov inequality to show that for every t > 0,P(X > a) ? e?taE[etX].(b) Suppose X is a Bernoulli random variable with parameter p. For every t > 0, find E[etX].(c) Suppose X1, X2,..., Xn are i.i.d. (independent and identically distributed) Bernoullirandom variables with parameter p. If S = X1 + ... + Xn, find E[etS].(d) Use the bound in part a to show that for every t > 0P(S > a) ? e?ta(etp + 1 ? p)n. (1)(e) (optional) The bound (1) hold for all t > 0. Which t gives the best bound? What isoptimal upper bound? (That is, minimize the upper bound with respect to t over t > 0

6. Suppose U1, U2, ... are i.i.d. random variables, each with uniform distribution over [0, 1]. Leth : [0, 1] 7? R is an integrable function, and define Xi = h(Ui) for i = 1, 2, .... Provelimn??X1 + X2 + ... + Xnn=Z 10h(x)dx,with probability one.

9. Suppose 60 percent of the people in a country support a law. A group of N people participatein a referendum about the law, and the law is passed if at lest half of the participants votein favor of the law. If the opinions are independent, what is the N for which the the law ispassed with probability 99.9%?

Math 180A - Homework 10

( Due Friday, Dec 2, 4:00 PM)

Reading: Sections 5.4.1, 6.3, 7.2, 7.4, and 8.1-8.3 of the textbook.

Show FULL JUSTIFICATION for all your answers. 1 Warm up Problems ? Do not turn in

1. Problems 7.6, 7.30 and 7.33 of the textbook.

2. Theoretical Exercises 7.1 and 7.2 of the textbook

3. Problem 8.1 of the textbook.

4. Theoretical Exercises 8.1 and 8.2 of the textbook. 2 Homework Problems ? To turn in

1. Suppose X is has a Normal distribution with mean ? and variance ? 2 .

(a) Define Y = ?X. Find the probability density function of Y .

(b) Given an example of two random variable with Normal distribution, such that their sum

does not have a Normal distribution.

2. Problem 7.7 of the textbook.

3. A group of 10 students volunteered to work abroad. Each student chooses a country at

random among 8 possible countries. If the choices are made independently, calculate the

expected number of countries which will host no volunteers.

4. Problem 7.12 of the textbook.

5. Fix a real number a.

(a) For any random variable X, use the Markov inequality to show that for every t > 0,

P (X > a) ? e?ta E[etX ].

(b) Suppose X is a Bernoulli random variable with parameter p. For every t > 0, find E[etX ].

(c) Suppose X1 , X2 ,..., Xn are i.i.d. (independent and identically distributed) Bernoulli

random variables with parameter p. If S = X1 + ... + Xn , find E[etS ].

(d) Use the bound in part a to show that for every t > 0

P (S > a) ? e?ta (etp + 1 ? p)n . (1) (e) (optional) The bound (1) hold for all t > 0. Which t gives the best bound? What is

optimal upper bound? (That is, minimize the upper bound with respect to t over t > 0.) 1 6. Suppose U1 , U2 , ... are i.i.d. random variables, each with uniform distribution over [0, 1]. Let

h : [0, 1] 7? R is an integrable function, and define Xi = h(Ui ) for i = 1, 2, .... Prove

X1 + X2 + ... + Xn

lim

=

n??

n Z 1 h(x)dx,

0 with probability one.

7. Problem 5.25 of the textbook.

8. Problem 8.4 of the textbook.

9. Suppose 60 percent of the people in a country support a law. A group of N people participate

in a referendum about the law, and the law is passed if at lest half of the participants vote

in favor of the law. If the opinions are independent, what is the N for which the the law is

passed with probability 99.9%? 3 More Practice ? Do not turn in

1. Problems 5.27 and 5.28 of the textbook.

2. Problems 7.8, 7.10, and 7.40 - 7.42 of the textbook.

3. Problems 8.2 and 8.3 of the textbook. 2

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