When calculating the convolution of two PDFs, one must be careful to use the appropriate limits of integration. Suppose that X and Y are nonnegative random variables.
In particular, fX(x) is equal to some positive function hX(x) for x >= 0 and is zero for x < 0.
Similarly, fY(y) is equal to some positive function hY(y) for y >= 0 and is zero for y <0.
Then the convolution integral is of the form:
\integrals_a^b hX(x) hY(z - x) dx
Find the values of a and b.
a =
b =
Certainly! To find the values of a and b for the convolution integral, we need to consider the limits of integration based on the nonnegative ranges of the random variables X and Y.
Let's start by looking at the limits for variable x. Since fX(x) is zero for x less than 0, we can set the lower limit of integration, a, to be 0. This ensures that we are only considering positive values of x.
For the upper limit of integration, b, we need to consider the range of possible values for x. In the convolution integral, we have the term (z - x) which represents the variable y in the function hY. Since Y is also nonnegative, we know that z - x must be greater than or equal to 0. Solving for x, we have x <= z.
Therefore, the upper limit of integration, b, can be set to z.
So, the values of a and b for the convolution integral are:
a = 0
b = z