How To Find Pmf From Cdf

How to Find the Probability Mass Function (PMF) from the Cumulative Distribution Function (CDF)

The probability mass function (PMF) and the cumulative distribution function (CDF) are two fundamental concepts in probability and statistics, particularly when dealing with discrete random variables. Understanding their relationship is crucial for analyzing data and making informed decisions. This article explains how to derive the PMF from the CDF, a process that's surprisingly straightforward. We'll cover the theoretical underpinnings and illustrate the process with practical examples.

What is a CDF?

The cumulative distribution function (CDF), denoted as F(x), gives the probability that a discrete random variable X will take a value less than or equal to x. In simpler terms, it's the cumulative probability up to a given point. For a discrete random variable, the CDF is a step function, meaning it increases in discrete jumps at each possible value of X.

What is a PMF?

The probability mass function (PMF), denoted as P(X=x) or p(x), gives the probability that a discrete random variable X will take on a specific value x. It represents the probability assigned to each individual outcome in the sample space.

Deriving the PMF from the CDF: The Key Relationship

The core relationship between the PMF and the CDF lies in the fact that the PMF represents the difference in the CDF between consecutive values. Mathematically, this can be expressed as:

P(X = x) = F(x) - F(x-1)

This formula holds true for all values of x within the range of the discrete random variable. Note that F(x-1) will be zero when x is the smallest possible value of the random variable.

Step-by-Step Guide to Finding the PMF from the CDF:

1. Identify the CDF: You'll need the CDF for the discrete random variable you're working with. This function will typically be given or derived from the problem statement.

2. Determine the Possible Values of X: List all the possible values that the discrete random variable X can take. This is crucial because you will calculate the PMF for each of these values.

3. Apply the Formula: For each possible value of x, substitute x and (x-1) into the formula: P(X = x) = F(x) - F(x-1). Remember that F(x-1) = 0 when x is the smallest possible value.

4. Verify the Results: The sum of all probabilities in the PMF should always equal 1. This is a crucial check to ensure the accuracy of your calculations. If the sum isn't 1, re-check your work.

Example:

Let's say we have the following CDF for a discrete random variable X:

• F(0) = 0.2
• F(1) = 0.6
• F(2) = 0.9
• F(3) = 1

Now, let's derive the PMF:

• P(X = 0) = F(0) - F(-1) = 0.2 - 0 = 0.2
• P(X = 1) = F(1) - F(0) = 0.6 - 0.2 = 0.4
• P(X = 2) = F(2) - F(1) = 0.9 - 0.6 = 0.3
• P(X = 3) = F(3) - F(2) = 1 - 0.9 = 0.1

Notice that the sum of all probabilities (0.2 + 0.4 + 0.3 + 0.1) equals 1, confirming the accuracy of our calculations. Therefore, we have successfully derived the PMF from the given CDF.

Conclusion:

Finding the PMF from the CDF is a fundamental skill in probability and statistics. By understanding the relationship between these two functions and following the steps outlined above, you can easily transition between them, enabling deeper insights into the behaviour of discrete random variables. Remember to always verify your results to ensure accuracy.