Question

Given
f(x) = (
c(x + 1) if 1 < x < 3
0 else
as a probability function for a continuous random variable; find
a. c.
b. The moment generating function MX(t).
c. Use MX(t) to find the variance and the standard deviation of X.

Answer 1

Given the PDF

f(x) =C(X + 1):1<x<3

a) The condition for valid PDF is

f(x) dx = 1 c/ (x + 1)dx = 1 C(4 + 2) = 1

b) The moment generating function (MGF) is

Mr (t) = E (e") My (t) = | etx f (x) dx e*dx 4e3tー2e 6t 6t2

c) The expectation is

0) - lim 6t2 6t2 6t3 6t2 6t3 6t2 2e)

Applying L'Hospital' rule,

018t +el) 108t2e3t 81te3t- 2t2e-5te Mx (0)-lim t-+0 36t t+0 36 19 E(X)9

Similarly,

0) - lim t+o dt 6t3 43 Mx (O)- 9 21 43 Var (X)-E (X2) -E(x) 19 26 81 Var (X) - /26 9