Question

12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)

Answer 1

Solution : la Let qke a continuous random variable with cumulative aistribution function ass Lo loq x xcal F(x) = as xc6 x 26 - ca We know thaus 2 F (a) = 0 s loglal=0 3 logcay = log (1) a=1 and F (6) = 1 log (by = loge b=& Hences a=1 and bie

P[x720) = 1 - P(X<2) = 1- Loga) Hence P(X72) = 6-log(2) © the probabilisep devsiz fanden f(2)=4 of x asy 1<x<e Olliagwise 1

ER)= ſufla, de Sulta da nd dx - Sade [ xje a e-1 Hence E(X)= e-l