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Q. 1 Suppose X and Y are independent discrete random variables having the Poisson distribution with parameters λ1 and λ2, respectively. Let Z = X +Y, Calculate E[Y | Z = z]. Q. 2 The random variables X and Y have joint probability density function fX,Y (x, y) = (cy x if 0 ≤ y < x ≤ 1, 0 otherwise

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EE 351K Probability, Statistics and Stochastic Processes – Spring 2016 Homework 5 Topics: Discrete and Continuous RVs Homework and Exam Grading “Philosophy” Answering a question is not just about getting the right answer, but also about communicating how you got there. This means that you should carefully define 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 practice thinking clearly and communicating your approach and/or ideas. To encourage you to be neat, if you homework or exam solutions are sloppy you may NOT get full credit even if your answer is correct. Q. 1 Suppose X and Y are independent discrete random variables having the Poisson distribution with parameters λ1 and λ2, respectively. Let Z = X +Y, Calculate E[Y | Z = z]. Q. 2 The random variables X and Y have joint probability density function fX,Y (x, y) = (cy x if 0 ≤ y < x ≤ 1, 0 otherwise. (a) Find the constant c, and the marginal densities fX (x) and fY (y). (b) Find P(X +Y ≤ 1). Q. 3 Suppose that the weight of a person selected at random from some population is normally distributed with parameters µ and σ 2 . Suppose also that P(X ≤ 160) = 1 2 and P(X ≤ 140) = 1 4 . Find µ and σ. Also, find P(X ≥ 200). (Use the standard normal table for this problem.) Q. 4 If X has the normal distribution with mean µ and variance σ 2 , find E X 3 (as a function of µ and σ 2 ), without computing any integrals. Q. 5 Let X and Y denote two points that are chosen randomly and independently from the interval [0,1]. Find the probability density function of Z = |X −Y|. Use this to calculate the mean distance between X and Y. Hint: It might be simpler to first calculate P(Z > z). Q. 6 Random variables X and Y are distributed according to the joint probability density function fX,Y (x, y) = ( C if x ≥ 0 and y ≥ 0 and x+y ≤ 1 0 otherwise. Let A be the event {Y ≤ 0.5} and B be the event {Y > X}. (a) Evaluate the constant C. (b) Calculate P(B | A). (c) Find E[X | Y = 0.5] and the conditional probability density function fX|B(x | B). (d) Calculate E[XY]. Q. 7 Let X and Y be independent random variables each uniformly distributed on [0,100]. Find the value of P(Y ≥ X | Y ≥ 12). Q. 8 A coin has an a priori probability X of coming up heads, where X is a random variable with probability density function fX (x) = ( xex , for x ∈ [0,1], 0, otherwise. (a) Find P(Head). Hint: note that according to the hypothesis P(Head | X = x) = x. (b) If A denotes the event that the last flip came up heads, find the conditional probability density function of X given A, i.e., fX|A(x | A). (c) Given A, find the conditional probability of heads at the next flip.

 
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