Question

) If the moment generating function (m.g.f.) of a random variable X is 

( ) exp 3( 32 ). 2 M t t t X = + Find mean and standard derivation of X and also compute 

P(x < )

Answer 1

Given mgf is-

$f(t)=e^{3t}$

First moment- Mean is given by

$E(X)=\int_0^1te^{3t}dt=t\dfrac{e^{3t}}{3}-\int_0^1\dfrac{e^{3t}}{3}dt$

$=\dfrac{2}{9}e^3-\dfrac{1}{9}$

Second moment -

$E(X^2)=\int_0^1t^2e^{3t}dt=t^2\dfrac{e^{3t}}{3}-\int_0^1\dfrac{2}{3}te^{3t}dt$

$=\dfrac{5}{27}e^3+\dfrac{2}{27}$

Standard deviation $=\sqrt{E(X^2)-[E(X)]^2}$

$=\sqrt{\dfrac{5}{27}e^3+\dfrac{2}{27}-\dfrac{4}{81}e^6-\dfrac{1}{81}+\dfrac{4}{81}e^3}$

$=\sqrt{\dfrac{19}{81}e^3-\dfrac{4}{81}e^6+\dfrac{5}{81}}$

P(X<a) can be computed by integrating f(t) from 0 to a.

$P(X<a)=\int_0^ae^{3t}dt=e^{3t}|_0^a=e^{3a}-1$