Jacobian transformation method to find marginal PDF of (X+Y)




It is always challenging to find the marginal probability density functions of several random variables like √X, (1/X), X+Y, XY, etc. when we know the marginal and/or joint probability density functions. We call these variables like √X, (1/X), X+Y, XY, etc. as transformations. Jacobian transformation method is one of the methods which is widely used to find marginal pdfs of such transformations. Let us understand how to use the Jacobian transformation method with the help of an example.

Question

Let X and Y be independent, continuous random variables with probability density functions (PDFs) as follows.


Use the Jacobian transformation method to calculate the PDF of X+Y.

Solution

We have two continuous random variables X and Y with their PDFs as follows.


Step 1: Obtaining joint probability distribution for (X, Y)

Two continuous random variables, X and Y, are independent if we can write the joint PDF of (X, Y) as a product of their marginal PDFs. In this case, we know that the random variables X and Y are independent. Therefore, the joint PDF of (X, Y) is given as follows.


Step 2: Obtaining range for newly defined variables

We need to find a marginal PDF of X+Y, for this, we define U=Y and V=X+Y. We will use the Jacobin transformation method to obtain the joint pdf of U, V. Then we try to define X and Y in terms of U and V as follows.


Recall the ranges of variables X and Y as, x ≥ 0, y ≥ 0. From these, we can easily obtain the ranges of random variables U and V as,



Step 3: Obtaining Jacobian (J)

The Jacobin | J | can be defined as follows,


From this we get,


This gives,



Step 4: Obtaining joint probability distribution for (U, V)

Therefore, we can obtain the joint pdf of (U, V) as follows.


This is the required joint PDF of (U, V). But we need to find the PDF of V=X+Y, for this, we proceed as follows.



Step 5: The marginal probability distribution of X+Y

Summarizing this, we get the marginal PDF of V = X+Y as follows.