# How to find the distribution for the following random variable $X$?

If $(X-20)\sim\operatorname{Poisson}(a)$, where $a$ is the model parameter for Poisson distribution. Then how is the random variable $X$ distributed? How to write down the pmf for random variable $X$?

Thanks a lot!

• If you have the probability mass function for $X-20$, can you find the one for $X$? Just add $20$ to the argument. – Jeppe Stig Nielsen Feb 22 '14 at 22:43

## 1 Answer

Using your notation, for Poisson$(a)$, $P(X=x)=\dfrac{a^x e^{-a}}{x!}$.

Then, for $X-20$, you have $P(X-20=x)=P(X=x+20)$. What do you think this will do to the pmf?