Transformation of Random Variables

Suppose X is a random variable with PDF f_X and CDF F_X. Let Y = r(X) be a function of X, then we call Y = r(X) a transformation of X.

In discrete case, the mass function of Y is given by
f_Y(y) = P(Y = y) = P(r(X) = y) = P({x; r(x) = y}) = P(X ∈ r^-1(y))

Example: Let us take the following mass function –

Let Y = X². Then, P(Y = 0) = P(X = 0) = 1/2 and P(Y = 1) = P(X = 1) + P(X = -1) = 1/2. Summarizing:

For continuous case, there are three steps for transformation
1. For each For each y, find the set A_y = {x : r(x) ≤ y}.
2. Find the CDF
                     F_y(y) = P(Y ≤ y) = P(r(X) ≤ y)
                               = P({x; r(x) ≤ y}) = ∫_Ay f_x(x)dx
3. The PDF is f_Y(y) = F’_Y(y).

Example: Let f_X(x) = e^−x for x > 0.
Hence, F_X(x) = ∫₀^x f_X(s)ds = 1 − e^−x …(i)
Let Y = r(X) = logX …(ii)
Then, A_y = {x : x ≤ e^y} and
            F_Y(y) = P(Y ≤ y) = P(logX ≤ y) …using eqn(ii)
            = P(X ≤ e^y) = F_X(e^y) = 1 − e^−ey … using eqn(i)
Therefore, f_Y(y) = e^ye^−ey for y ∈ R.