Question

7.4 Let X ~ U(-1,1) and Y = x2. a. What are the density, the distribution function, the mean, and the variance of Y: b. What is Pr[Y < 0.5]? 7.5 Let X – U(0,1), and let Y = eax for some a > 0. What are the density, the distribution function, the mean, and the variance of Y?

Answer 1

releri Givent- - XNufar, 1) and y=x2 a The probability density function (ppp) of random variable 'x'ik given by fx Ca): Jā if-14851 otherwire 101 it is given that y=x² the PIDE of random variable "y' is obtained by the formula ! fyCy) = ready the cas e S - Vy af & zos azavy

5/ = (h) (-34-I CADA Por el be hroty z = C-D2 =()2 = crow 3 rool - solo est. et al y dy maan = ECO: (v pondye li na tayo ples 1 * +44 1 then A TERMITTIT E TIT

Variance () = EC42) - (49]? 5 § - Cu) wariance (1 = do variance (Y) =0.0891 50.089 o probability dwadzibution frenetton ive indicated by Fy Cy) and Fycy) = J'Arcoody sto ( of 'Y' limits Slants from zero) v arg sto * x* [yX_0] | FyCD = y Y Pr[1905] = 0.5942-0 = 0.707 Pr[780.5] = 0.7077