Fall 2011 Strategic Practice 8: Section 2 (Transformations) - Question 3
Let X and Y be independent, continuous r.v.s with PDFs $f_{X}$ and $f_{Y}$ respectively, and let T = X + Y . Find the joint PDF of T and X, and use this to give an alternative proof that $f_{T}(t)=\int_{-\infty }^{\infty }f_X{x}f_{Y}(t-x)dx$, a result obtained in class using the law of total probability.
Solution: Consider the transformation from (x, y) to (t, w) given by t = x + y and w = x. (It may seem redundant to make up the new name "w" for x, but this makes it easier to distinguish between the "old" variables x, y and the "new" variables t,w.) Correspondingly, consider the transformation from (X, Y) to (T, W) given by T = X + Y, W = X. Use the Jacobian to find the joint PDF of T, W and integrate out W to get the marginal PDF of T.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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