How Do You Find the Marginal Density Function?

The marginal density function of a random variable is found by integrating the joint density function over all possible values of the other variable(s).

Here's a breakdown:

Concept:

Imagine you have a joint probability density function (PDF) that describes the probability of two or more random variables taking on certain values simultaneously. The marginal density function tells you the probability distribution of just one of those variables, disregarding the others. In essence, you're "averaging out" the effect of the other variables.

Calculation:

For two random variables, X and Y, with a joint PDF f(x, y), the marginal PDFs are calculated as follows:

• Marginal PDF of X: Integrate the joint PDF with respect to y over its entire range:

  f_X(x) = ∫ f(x, y) dy

• Marginal PDF of Y: Integrate the joint PDF with respect to x over its entire range:

  f_Y(y) = ∫ f(x, y) dx

Generalization:

If you have more than two variables, say X, Y, and Z, the marginal PDF of X is found by integrating the joint PDF over all possible values of both Y and Z:

f_X(x) = ∫∫ f(x, y, z) dy dz

Example:

Let's consider an example similar to the one mentioned in your references (although the specific joint PDF in your reference snippet is incomplete). Suppose the joint PDF of X and Y is given by:

f(x, y) = C, for -1 ≤ x ≤ 1 and -√(1-x²) ≤ y ≤ √(1-x²)
f(x, y) = 0, otherwise.

Where C is a constant. This represents a uniform distribution within a circle of radius 1. To find C, the double integral of f(x,y) over the region must equal 1. Integrating this gives an area of pi, so C = 1/pi.

To find the marginal PDF of X, f_X(x), we integrate f(x, y) with respect to y:

f_X(x) = ∫_-√(1-x²)^√(1-x2) (1/π) dy = (1/π) [y]_-√(1-x²)^√(1-x2) = (1/π) [√(1-x²) - (-√(1-x²))] = (2/π)√(1-x²)

So, f_X(x) = (2/π)√(1-x²), for -1 ≤ x ≤ 1 and 0 otherwise.

Key Points:

• The limits of integration are determined by the range of the variable you are integrating over, which is defined by the joint PDF.
• The resulting marginal PDF is a function of only the variable you didn't integrate over.
• The marginal PDF must integrate to 1 over its entire range.

In summary, calculating the marginal density function is a fundamental operation in probability and statistics, allowing you to analyze the distribution of individual variables within a joint distribution.