Final answer:
To determine the pdfs of various combinations of two independent random variables x and y, we can use the convolution theorem to find the pdf of x + y, x - y, xy, and x/y. The pdfs of these combinations can be calculated by multiplying the individual pdfs of x and y.
Step-by-step explanation:
To determine the probability density functions (pdf) of various combinations of two independent random variables x and y, we need the pdf for the individual variables, which is given as g. Let's denote the pdf of x + y as h. To find h, we can use the convolution theorem, which states that the pdf of the sum of two independent random variables is the convolution of their respective pdfs.
Applying the convolution theorem, we have h = f(x) * f(y), where * denotes the convolution operation. Since x and y are independent, their joint density function is the product of their individual density functions: g(x, y) = g(x) * g(y).
Similarly, we can find the pdfs for the other combinations:
b. The pdf of x - y is g(x) * g(-y), where g(-y) is the pdf of the variable -y.
c. The pdf of xy is g(x) * g(y).
d. The pdf of x/y is g(x) * g(1/y), where g(1/y) is the pdf of the variable 1/y.