Answer to Question #203986 in Statistics and Probability for Vaishu

Question #203986

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


1
Expert's answer
2021-06-08T12:15:27-0400

E(X)=dMXdtt=0=(3+64t)e3t+32t2t=0=3E(X)=\frac{dM_X}{dt}|_{t=0}=(3+64t)e^{3t+32t^2}|_{t=0}=3


E(X2)=d2MXdt2t=0=64e3t+32t2+(3+64t)2e3t+32t2t=0=67E(X^2)=\frac{d^2M_X}{dt^2}|_{t=0}=64e^{3t+32t^2}+(3+64t)^2e^{3t+32t^2}|_{t=0}=67


σ=V(X)=E(X2)(E(X))2=679=7.6\sigma=\sqrt{V(X)}=\sqrt{E(X^2)-(E(X))^2}=\sqrt{67-9}=7.6


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


P(X3)=e3te3t+32t2=e32t2P(X\ge 3)=e^{-3t}e^{3t+32t^2}=e^{32t^2}


P(X<3)=1P(X3)=1e32t2P(X<3)=1-P(X\ge 3)=1-e^{32t^2}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment