< Probability distributions < 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 :
- Find the cumulative distribution function (CDF) FY(y) = P(Y ≤ y),
- Differentiate the CDF to find the pdf. In notation, fY(y) = FY′(y).
Method of distribution functions example
Example 1: Suppose that X is a continuous random variable with pdf fX(x) = 1 / (1 + x) on the interval (0, e – 1). If another random variable Y is defined in terms of X as
Y = ½ √ (1 + X),
what is fY(y)?
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:
- FX(x) = ln | 1 + x | + C
- ln | 1 + 0 | + C
- 0 = ln(1) + C
- C = 0.
Therefore, the CDF is FX(x) = ln(1 + x) on (0, e – 1).
Two more examples (video):
Other methods to find pdfs of functions of random variables include :
- 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.
 Oenn 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