Exercise 3.1. a) Let X, Y be III rv's, each with density fx (x)= a exp(-1/2)....

  

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Exercise 3.1. a) Let X, Y be III rv's, each with density fx (x)= a exp(-1²/2). In (b), we show that a must be 1/√2 in order for fx(r) to integrate to 1, but in this part, we leave a undetermined. Let S = X² + y2. Find the probability density of S in terms of a. b) Prove from (a) that a must be 1/√2 in order for S, and thus X and Y, to be random variables. Show that E[X] = 0 and that E [X²] = 1. c) Find the probability density of R= √5. R is called a Rayleigh rv. Exercise 3.1. a) Let X, Y be III rv's, each with density fx (x)= a exp(-1²/2). In (b), we show that a must be 1/√2 in order for fx(r) to integrate to 1, but in this part, we leave a undetermined. Let S = X² + y2. Find the probability density of S in terms of a. b) Prove from (a) that a must be 1/√2 in order for S, and thus X and Y, to be random variables. Show that E[X] = 0 and that E [X²] = 1. c) Find the probability density of R= √5. R is called a Rayleigh rv.

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