Question

let x be a random variable which takes values in the interval (1,3). the density function of x is proportional to 2^x. find the mean

Answer 1

TOPIC:Expectation of random variables.

Define blf, dex). the nandom variable X, with So, Hx) is proopontional. to 2 . fa) x 2 where, Hx) = k.2 k= constant. So, we have, j dor, 1x 3 k. 2 ; otherulise. values X takes within the interval (,3) But, Hx) being must have, a valid bof j we 3. Hx) dx =1. A k. 27

3 In 2 In b 3 =) 21. 2. In 2 : The p4 of x becomes In 2 dex) = ; Jon, 1<x <3. 2. j otheruise , , we meed, mean(x). 3. X Jex) dx. In 2 X 2 dx,

(By Integration by parts 6. In 2 dz In 2 In 5 In 2. 3.2 - 1.2 In 2 In 2 3. 22 1n 2 In 2 In 2 22 In 2 22 In 2-6 6. In 2 1l In 2-3 3. In 2 2.2239716 (An).

of Fornmula integrmation by parts- u (x).v (x)dx. fucomJor Taking, u(a) = X, and, v(x) = 2