# Probability Density Function of the Product of Independent Variables

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• megf

#### megf

TL;DR Summary
How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b?
How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which functions should i use for the product?

Let the random variables be ##A## and ##B##, with density functions ##f_A## and ##f_B## respectively. Let ##Y=AB##. For simplicity, assume that ##A,B## can only be positive.
1. express the joint probability density function ##f_{A,B}## of the two variables in terms of ##f_A## and ##f_B##. Use the fact that they are independent.

2. calculate the cumulative distribution function of Y:
$$F_Y(y) = \int_{0}^\infty \int_{0}^{y/a} f_{A,B}(a,b)\, db\,da$$

3. Differentiate ##F_Y## to get ##f_Y##, the PDF of ##Y##.

If A or B can be zero, the solution needs to be more complex, to avoid divisions by zero. The integrations need to be split into parts that avoid zero.

By the way, the formula you provided for the PDF of A/B does not look right, as it contains no integrals. Have you omitted some info from the question, such as that the two variables are standard normal, or some other specific distribution?

Let the random variables be ##A## and ##B##, with density functions ##f_A## and ##f_B## respectively. Let ##Y=AB##. For simplicity, assume that ##A,B## can only be positive.
1. express the joint probability density function ##f_{A,B}## of the two variables in terms of ##f_A## and ##f_B##. Use the fact that they are independent.

2. calculate the cumulative distribution function of Y:
$$F_Y(y) = \int_{0}^\infty \int_{0}^{y/a} f_{A,B}(a,b)\, db\,da$$

3. Differentiate ##F_Y## to get ##f_Y##, the PDF of ##Y##.

If A or B can be zero, the solution needs to be more complex, to avoid divisions by zero. The integrations need to be split into parts that avoid zero.

By the way, the formula you provided for the PDF of A/B does not look right, as it contains no integrals. Have you omitted some info from the question, such as that the two variables are standard normal, or some other specific distribution?

Oh, yes, they were standard normal in the A/B example I mentioned. My bad. Though I'm not sure if I can say my variables are standard normal variables.

Summary: How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b?

How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which functions should i use for the product?
If A, B are standard normals A/B has a Cauchy distribution.