Question

Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)

Answer 1

S ay xa Exp (2=1 fla= e x >0 P(x731871) 1 - P(x > 3, 871) P(x>1) C8x73} - P(X73) as d x > P(x71) 1-P(X53) - PlX =) Javo) F1*2=R-e ;x? to 2 jow 1-1x13) infix (1) 1-(-ė) SON 23 10 = 0.135335 1-(-e'l (approx) ay Amy 0.13 5335 F1: 20

b> YN (4,6²) 3.1 M) fy 14) Jero e y ER M E IR A OA ; o.w . is pdf of Y. 2 Yan 14,8²) Z=1-M N (01) P(Z <=) = D(z); ZEIR P (Y7M+36) = 1-PLYL M+38) > RC = 1-PCV-A 436-) = |- P(Z < 3) = 1- $(3) = 1-0.99865 -0.00135 18 (z) values are obtained from Standard Normal Table or Calculator of