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Math 426 Quiz 5 1. Consider two discrete r.v. X and Y with joint pmf given by 1 f (10, 1) = f (20, 1) = f (20, 2) = 10 , f (10, 2) = f (10, 3) = 15 , f (20, 3) = Are X and Y independent? The joint and marginal probabilities are given in below. 3 10 . y 1 2 3 f(x) 10 1/10 2/10 2/10 1/2 20 1/10 1/10 3/10 1/2 f(y) 1/5 3/10 1/2 x For the pair (10, 2), we have f (10, 2) = 0.2 6= fX (10) ∗ fY (2) = 0.5 ∗ 0.3 = 0.15. So X and Y are not independent. 2. Suppose a continue r.v. has a normal distribution with pdf fX (x) = 2 2 1 √ e−(x−µ) /2σ , −∞ < x < ∞. Find the pdf of Y = eX . 2πσ Note y = ex is an increasing function for −∞ < x < ∞. 1 From y = ex , we get x = ln y, and dx dy = y . So the pdf of Y is given by 2 2 2 2 1 1 fY (y) = √2πσ e−(lny−µ) /2σ y1 = √2πσy e−(lny−µ) /2σ , y > 0. Y is said to have a log-normal distribution because the logarithm of Y has a normal distribution. 2 3. Consider the r.v. X with pdf fX (x) = 3x2 , −1 ≤ x ≤ 1. Find the pdf of Y = |X|. Hint: use the cdf technique. First note y = |x| is not a monotone function of x on −1 ≤ x ≤ 1. Ry 2 FY (y) = P (Y ≤ y) = P (|X| ≤ y) = P (−y ≤ X ≤ y) = −y 3x2 dx = x3 y 2 |−y = y 3 , so fY (y) = 3y 2 , 0 ≤ y ≤ 1. 1