Transformations of Random Variables
Some hints on figuring out the distribution of the transformation of a random variable, Y=g(X).
- Figure out the support of Y (the set of possible values Y can take on).
- Is the transformation strictly monotone? Then use the rules for strictly monotone transformations.
- If that doesn't work, try to calculate the CDF. It may help to divide the set of possible y values into different subintervals.
Calculating the expectation of a transformation of a random variable: Let `Y=r(X)`. Then (in the discrete case)
`E[r(X)] = sum_x r(x) f_X(x) = sum_y y f_Y(y).`
In class, we saw a simple proof for the case where the transformation `r` is strictly monotone. Actually, it's easy to extend this to any transformation. This proof can be found in DeGroot's textbook (see syllabus):
`sum_y y f_Y(y) = sum_y y P[r(X)=y] = sum_y y sum_{x:r(x)=y} f_X(x) `
because the probability that `r(X)=y` is just the sum of the probabilities of the different `x` values such that `r(x)=y`. This then equals
`sum_y sum_{x:r(x)=y} r(x) f_X(x) = sum_x r(x) f_X(x).`
As mentioned in class, the continuous case also works. I (Kei) only know of a fairly complicated proof, unfortunately.