Method of distribution functions

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What is the method of distribution functions?

Given a distribution of a random variable X, the method of distribution functions gives the probability distribution of U = f(X). It works when X is a continuous random variable.

In general, you can use the method of distribution functions to find the probability density function (pdf) of a random variable Y = μ(X) by following these steps [1]:

• Find the cumulative distribution function (CDF) F_Y(y) = P(Y ≤ y),
• Differentiate the CDF to find the pdf. In notation, f_Y(y) = F_Y′(y).

Method of distribution functions example

Example 1: Suppose that X is a continuous random variable with pdf f_X(x) = 1 / (1 + x) on the open interval (0, e – 1). If another random variable Y is defined in terms of X as

Y = ½ √ (1 + X),

what is f_Y(y)?

Solution:

Step 1: Find the antiderivative of the pdf of X to get the CDF. I used Symbolab to get this solution:

Step 2: Find the constant C (for the CDF in step 1). To do this, use one of the CDF bounds. We know the bounds are (0, e – 1) from the question. The first value, 0, will be easier to work with than e – 1, so we’ll plug that into F(x) = ln |1 + x| to get C:

• F_X(x) = ln | 1 + x | + C
• ln | 1 + 0 | + C
• 0 = ln(1) + C
• C = 0.

Therefore, the CDF is F_X(x) = ln(1 + x) on (0, e – 1).

Two more examples (video):

 

Similar methods

Other methods to find pdfs of functions of random variables include [2]:

• Method of (univariate) transformations: the density of Y determines density of U = f(Y) directly, without going through the distributions. The method requires the function U = f(Y) to be continuous and one-to-one monotonic.
• Method of moment–generating functions: Given the moment generating function of Y (mgf), this method gives the mgf of U = f(Y). This method is particularly useful if the probability distributions/densities have a recognizable moment-generating functionsmgf.

References

[1] Penn State, Eberly College of Science. 22.1 – Distribution Function Technique. STAT 414: Introduction to probability theory. Retrieved October 21, 2023 from: https://online.stat.psu.edu/stat414/lesson/22/22.1

[2] Chapter 6. Functions of Random Variables (lecture notes).