Question

the mgf of a random variable x is e^3(e^t-1). Find P[mean - standard deviation squared < X < 1/2( mean + standard deviation squared)], where mean and standard deviation squared are mean and variance of X

Answer 1

The given MGF is that for a random variable following a Poisson distribution with parameter $\lambda=3$.

This means $\mathbb E(X)=\mathbb V(X)=\lambda$, and $X$ has PMF

$f_X(x)=\begin{cases}\dfrac{3^xe^{-3}}{x!}&\text{for }x\ge0\\0&\text{otherwise}\end{cases}$

So, the desired probability is

$\mathbb P\left(\lambda-\lambda^2$

This is equivalent to

$\displaystyle\sum_{x=0}^2\mathbb P(X=x)=\sum_{x=0}^2\frac{3^x}{x!e^3}=\frac{17}{2e^3}\approx0.4232$