Question

Let \(X\) have the pmf $$ p(x)=\left(\frac{1}{2}\right)^{|x|}, \quad x=-1,-2,-3, \ldots $$ Find the pmf of \(Y=X^{4}\).

Short answer

The probability mass function of \(Y = X^4\) is \(P(Y = y) = 2\left(\frac{1}{2}\right)^{\sqrt[4]{y}}\), for \(y = 1, 4, 9, ...\).

Step-by-step solution

Identify the values that Y can take

Since \(Y = X^4\), and \(X\) can take values of -1, -2, -3, ..., we have that \(Y\) will take on the values of 1, 4, 9, ... \(\infty\).

Determine the pmf of Y

The pmf of \(Y\) will be the sum of the probabilities of the values \(X\) that yield the value \(y\). For each value \(y\) that \(Y\) can take, we add the probability of \(X = \sqrt[4]{y}\) and \(X = -\sqrt[4]{y}\). The pmf is \(P(Y = y) = P(X = \sqrt[4]{y}) + P(X = -\sqrt[4]{y}) = \left(\frac{1}{2}\right)^{\sqrt[4]{y}} + \left(\frac{1}{2}\right)^{-\sqrt[4]{y}} = 2\left(\frac{1}{2}\right)^{\sqrt[4]{y}}\).