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}(3^xe^(-3))/(x!)&\text{for }x\ge0\\0&\text{otherwise}\end{cases}`

So, the desired probability is


`\mathbb P\left(\lambda-\lambda^2<X<\frac12(\lambda+\lambda^2)\right)=\mathbb P(0<X<\lambda)=\mathbb P(X<3)`

This is equivalent to


`\displaystyle\sum_(x=0)^2\mathbb P(X=x)=\sum_(x=0)^2(3^x)/(x!e^3)=(17)/(2e^3)\approx0.4232`