I had a homework problem with a question and solution as below.

Question: Let X be geometrically distributed with parameter p, let Y = $X^4$. Find the probability mass functions of Y.

My solution: $P(X=k) = (1-p)^{k-1} * p. P(Y = k) = P(X^4 = k) = P(X = k^{1/4}) = (1-p)^{x^{1/4} – 1} * p $.

The solution is: By definition, the probability mass function of X is given by

$P(X=k) = (1-p)^{k-1} * p$ â k = 1, 2, 3, . . .

Therefore the only values that Y takes with non-zero probability are 4th powers of positive integers, and $P(Y = k^4) = (1 â p)^{kâ1}p$ â k = 1, 2, 3, . . .

I’m confused by why my solution isn’t the correct approach and why Y could be directly perceived as such. It seems that I have a lack of understanding of how PMFs work.

Thanks!