# Probability Density Function of the Product of Independent Variables

• I
• megf
In summary: If A and B are not standard normals the integrals in the Cauchy distribution will produce a division by zero error.
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?

andrewkirk said:
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.

megf said:
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.

## 1. What is a Probability Density Function (PDF)?

A Probability Density Function (PDF) is a mathematical function that describes the probability distribution of a continuous random variable. It shows the relative likelihood of a random variable taking on a particular value or range of values.

## 2. What does it mean for variables to be independent?

Independent variables are variables that do not affect each other. In other words, the value of one variable does not impact the value of the other variable. In probability, this means that the outcome of one event does not affect the outcome of the other event.

## 3. How is the PDF of the product of independent variables calculated?

The PDF of the product of independent variables is calculated by multiplying the individual PDFs of each variable. This is based on the principle that the joint probability of independent events is equal to the product of their individual probabilities.

## 4. What is the significance of the PDF of the product of independent variables?

The PDF of the product of independent variables is important in probability and statistics because it allows us to calculate the probability of multiple independent events occurring simultaneously. This can be useful in various fields such as finance, engineering, and science.

## 5. Can the PDF of the product of independent variables be used to calculate the probability of dependent events?

No, the PDF of the product of independent variables can only be used for independent events. For dependent events, a different approach, such as using a joint probability distribution, would be needed to calculate the probability.

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