There are random variable $Z=XY$ ($X$ is poisson and $Y$ is bernoulli)
$$X(n;\lambda) = \frac{\lambda^n}{n!}e^{-\lambda}$$
$$Y=\begin{cases} & \beta \text{ with probability } \beta \\ & 0 \text{ probability } 1-\beta \end{cases}$$
I likes to know distribution of product of bernoulli random variable and poisson random variable.
So i calculate MGF(Moment Generating Function) so i can get below expression.
$$M_{z}(t)=\sum_{y}\sum_{x}e^{txy}P(x)P(y)=\sum_{x}(\beta e^{t\beta x}+1-\beta) P(x)
=\beta e^{-\lambda}(e^{\lambda e^{t\beta}})+1-\beta = \beta(e^{\lambda(e^{t\beta -1})}+1-\beta)$$
I can`t obtain pmf(probability mass function) from this calcultaed MGF
Is it impossible obtain pmf ?
Or is there any technique obtaining probability in this case?
Thank you
Best Answer
You can think about it for a bit. Presumably theres a typo in $Y$, which should be 1 w.p. $\beta$.
Note that $Z$ is zero iff $Y=0$ or $X=0$ (which are independent events), so $P(Z=0) = P(Y=0) + P(X=0) - P(X=0)P(Y=0)$.
Note that $Z=z$ where $z$ is a positive integer if and only if $Y=1$ and $X=z$ (which are independent events), so $P(Z=z) = P(Y=1) P(X=z)$.