Question

if the moment generating function (m.g.f) of a random variable X is M_x (t) = exp(3t + 32t²) . find mean and standard deviation of X and also compute P(x<3)

Answer 1

$E(X)=\frac{dM_X}{dt}|_{t=0}=(3+64t)e^{3t+32t^2}|_{t=0}=3$

$E(X^2)=\frac{d^2M_X}{dt^2}|_{t=0}=64e^{3t+32t^2}+(3+64t)^2e^{3t+32t^2}|_{t=0}=67$

$\sigma=\sqrt{V(X)}=\sqrt{E(X^2)-(E(X))^2}=\sqrt{67-9}=7.6$

$P(X\ge a)=e^{-at}M_X(t)$

$P(X\ge 3)=e^{-3t}e^{3t+32t^2}=e^{32t^2}$

$P(X<3)=1-P(X\ge 3)=1-e^{32t^2}$