Transformations of Random Variables

Often you know the distribution of $X$, but you care about a transformed variable:

There are two standard tools:

1. CDF method (works for discrete or continuous; especially good for monotone transforms)
2. Change of variables / Jacobian (continuous PDFs)

CDF method (general)

For any $y$:

If you can rewrite the event $g(X)\le y$ in terms of $X$, you can compute $F_Y(y)$, then differentiate (continuous case) to get $f_Y(y)$.

Continuous change-of-variables (monotone)

If $Y=g(X)$ is strictly monotone and differentiable, and $X$ has density $f_X$:

Linear transforms

If $Y=aX+b$:

• $E[Y]=aE[X]+b$
• $\mathrm{Var}(Y)=a^2\mathrm{Var}(X)$
• If $X$ is Normal, then $Y$ is also Normal.

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