Distributions of $X^2$ and $X-1$ when $X$ is geometric

Question

Let$ X$ be a discrete random variable with the probability mass function given by $p_x(x)= 2^{-x}$ for $x=1,2,3,\ldots$ and $0$ otherwise.

a) Let $Y=X^2$, find the probability mass function of $Y$;

b) Let $Z=X-1$, find the probability mass function of $Z$, identify the name of the distribution of $Z$ and all its parameter values.

So the probability mass function is the density function for discrete case right? I tried to integrate the function but somehow got a weird one. Is my thinking correct? thanks for any help!

Answer 1

Hint: if $g$ is a function and $X$ has probability distribution $p_X$, then

$$E[g(X)]:=\sum_{x}p_X(x)g(x),$$

where the sum is over all $x=1,2,3,\dots$, as stated in the OP.

You can find the $g$'s which define $Y$ and $Z$ and compute the corresponding expectation values. On the distribution of $Y$; as the image of $Y$ is the set of all integers $\{x^2, x=1,2,3,\dots\}$, then $p_Y(x^2)=2^{-x}$, which follows by using the expectation value formula above.