Final answer:
The density of the sum W = X + Y, where X and Y are independent random variables with given densities, is found by convolving their density functions. The convolution integral will give the probability density function (pdf) for W.
Step-by-step explanation:
To find the density of the sum W = X + Y, where X and Y are statistically independent random variables with given densities, we need to use the convolution of the two density functions. The respective densities are given as:
- fx(x) = 5u(x) exp(-5x)
- fy(y) = 2u(y) exp(-2y)
Since the variables are independent, the probability density function (pdf) of the sum W can be found by convolving the pdfs of X and Y:
fW(w) = ∫ fX(x) * fY(w - x) dx
This integral computes the area of the product of the two density functions over all values of x, which yields the density function of their sum, W.
Executing the convolution, we'll get:
fW(w) = ∫ 5u(x) exp(-5x) * 2u(w - x) exp(-2(w - x)) dx
This results in a new function that will describe the probability density function for the sum W. Calculating this integral, considering the support u (indicator function) of each random variable, results in the final density function for W.