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(Solved) EE 351K Probability, Statistics and Stochastic Processes - Spring

Please complete the assignment attached below and provide neat explanations. Thank you.?

The topics in this assignment include:?Random Sums, Moment Generating Functions, Bivariate Normal, Inequalities

EE 351K Probability, Statistics and Stochastic Processes - Spring 2016

Homework 8

Topics: Random Sums, Moment Generating Functions, Bivariate Normal, Inequalities

Homework and Exam Grading ?Philosophy?

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that you should carefully de?ne your model and notation, and provide a step-by-step explanation on how you got to your answer.

Communicating clearly also means your homework should be neat. Take pride in your work it will be appreciated, and you will get

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solutions are sloppy you may NOT get full credit even if your answer is correct.

Q. 1 A fair coin is thrown 100 times. Let X be the number of heads obtained.

(a) Using Markov?s inequality ?nd an upper bound on P(X > 70).

(b) Using Chebyshev?s inequality ?nd a lower bound on P(30 ? X ? 70).

(c) Using your answer to Part (b) and the symmetry of the distribution of X, obtain a better upper bound on P(X > 70).

Q. 2 Let U and V be independent standard normal random variables, i.e., N(0, 1), and let X = U + V and Y = U ? 2V . Find

fX|Y (x | y), E[X | Y = y], Cov(X,Y ), and the joint probability density function of X and Y .

Q. 3 Recall that the sum of a ?xed number of independent Gaussian random variables is Gaussian. Recall also that X is a Gaussian

?2 s2

random variable having zero mean if and only if its MGF has the form MX (s) = e 2 , where ?2 is the variance of X. This exercise

investigates whether a random sum of independent Gaussian random variables may also be Gaussian. Suppose that N is a random

variable taking values 1 and 3, each with probability 1 , and Xi , i = 1, 2, . . . , are independent and identically distributed Gaussian

2

random variables with mean 0 and variance 1, and also independent from N. Let WN = ?N Xi . Is WN a Gaussian random variable?

i=1

Try to ?nd a discrete random variable N taking positive integer values such that WN is Gaussian. Note then that we must have

E e

Ns2

2

= eas

2

for some positive constant a. If this equation holds, what should the value of the constant a be? What kind of random variables N

can match this?

Q. 4 Suppose that the continuous random variable X has PDF given by

1

fX (x) = e?|x| ,

2

?? < x < ?.

(a) Obtain the MGF of X.

(b) Using the MGF, ?nd E[X] and Var(X).

Q. 5 At a certain time, the number of people that enter an elevator is a Poisson random variable with parameter ?. The weight of

each person is independent of other person?s weight, and is uniformly distributed between 100 and 200 lbs. Let Xi be the fraction of

100 by which the ith person exceeds 100 lbs, e.g., if the 7th person weighs 175 lbs, then X7 = 0.75. Let Y be the sum of the Xi .

(a) Find MY (s).

(b) Use MY (s) to ?nd E[Y ].

(c) Verify your answer to part (b) by using the law of iterated expectations.

1

Q. 6 Suppose that Xi , i = 1, . . . , 100, are random variables measuring the annual rainfall (in inches) in a part of the Sahara dessert,

over 100 consecutive years. Assume that these are independent and identically distributed, each having a mean of 1 inch and a

standard deviation also of 1 inch.

(a) First assume that the random variables Xi are Gaussian (ignoring the fact that rainfall cannot be negative). Find the probability

that the total rainfall over the 100 year period exceeds 120 inches.

(b) Suppose you do not know the underlying distributions of the random variables. Using Chebyshev?s inequality, determine a

bound on the probability that the total rainfall exceeds 120 inches.

Q. 7 Every student applicant at a certain university is declined with probability 0.3. In other words, each decision to admit can be

modeled as a Bernoulli random variable. Assume that these decisions are independent. If the number of applicants in 2016 has the

Poisson distribution, with parameter 900, use the MGF to ?nd the probability distribution of the number of admitted students. If the

probability that a student who starts his/her studies in 2016 manages to graduate is 0.9 (assume independence), what is the expected

s

number of students that graduate? (Hint: if N is Poisson with parameter ?, then MN (s) = e?(e ?1) .)

2

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