Answer Happy

1. Let the random variable X have the Laplace distribution with pdf fx(x)=b exp(-2b|xl), for b>0 and corresponding cumulative distribution function (cdf) given by 2bx x < 0 Fx(x) = x>0 (a) Find the mean, median, mode, and standard deviation of X. (b) Give an algorithm to generate the random variable X. (c) Find the pdf of Y = X² and give an algorithm to generate it. Fxy (x, y) = -2bx 2. (a) Let the random variables X and Y have the joint distribution 0, x <0 or y < 0 0≤x≤4 45 x and any y 20 5/x+e-(x+1)y² X+1 -e-x²), (i) Find the marginal distributions Fx (x) and Fy (y). (ii) Compute the joint probability P(3 < X < 5,1 <Y ≤ 2). (iii) Compute the conditional probability P(3 < X < 51 <Y ≤ 2) (b) Let X and Y be independent and identically distributed random variables with mean μ and variance o². Find the following: (i) E[(X-Y)²] (ii) E[(X+Y)²] (iii) Cov{(X+Y), (X - Y)}