3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for...

Question

Question

3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e

Answer 1

Answer #1

Markov inequality states that, if Y is a non negative random variable,

for a>0,

$P(Y>a)\leq \frac{E(Y)}{a}$

Note that, $e^{tX}$ is non negative.

So,

$P(X>1)=P(tX>t)=P(e^{tX}>e^t)\leq \frac{E(e^{tX})}{e^t}$ (by markov inequality)

where, $Y=e^{tX}, a=t, M_X(t)=E(e^{tX})$

So,

Mx t) P(X > 1) P(X >1) 3 Mx(t)e

Answer 2

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