How To Find Pmf From Cdf

How to Find PMF from CDF: A Comprehensive Guide

The Probability Mass Function (PMF) and the Cumulative Distribution Function (CDF) are fundamental concepts in probability and statistics, particularly when dealing with discrete random variables. Understanding their relationship and how to derive one from the other is crucial for various applications. This comprehensive guide will walk you through the process of finding the PMF from the CDF, explaining the underlying theory and providing practical examples.

Understanding PMF and CDF

Before diving into the conversion process, let's refresh our understanding of PMF and CDF:

Probability Mass Function (PMF)

The PMF, denoted as P(X=x), gives the probability that a discrete random variable X takes on a specific value x. It's defined only for discrete values of x, and the sum of probabilities for all possible values of x must equal 1. Formally:

• P(X=x) ≥ 0 for all x (Probabilities are non-negative)
• ∑ P(X=x) = 1 (The sum of all probabilities is 1)

Cumulative Distribution Function (CDF)

The CDF, denoted as F(x), gives the probability that a random variable X takes on a value less than or equal to x. It's defined for all values of x, whether discrete or continuous. For a discrete random variable:

• F(x) = P(X ≤ x) = ∑ P(X=k) for all k ≤ x

The CDF is a non-decreasing function, meaning F(x) ≤ F(y) if x ≤ y. It approaches 0 as x approaches negative infinity and approaches 1 as x approaches positive infinity.

Deriving PMF from CDF: The Process

The key to obtaining the PMF from the CDF lies in understanding the relationship between the two functions. The CDF represents the accumulated probability up to a given point, while the PMF represents the probability at a single point. Therefore, to find the PMF, we need to isolate the probability at each discrete point.

Here's the step-by-step process:

1. Identify the Discrete Values: First, determine all the possible discrete values that the random variable X can take. These values will form the domain of your PMF.

2. Calculate the Probability at Each Point: For each discrete value xᵢ, the probability P(X=xᵢ) can be calculated using the following formula:

   • P(X=xᵢ) = F(xᵢ) - F(xᵢ₋₁), where F(xᵢ) is the CDF at xᵢ, and F(xᵢ₋₁) is the CDF at the value immediately preceding xᵢ. For the smallest value of x, F(xᵢ₋₁) = 0.

3. Verify the Sum: As a final check, ensure that the sum of all probabilities calculated in step 2 equals 1. If it doesn't, there's an error in your calculations.

Examples: Deriving PMF from CDF

Let's illustrate the process with several examples, progressing in complexity:

Example 1: Simple Discrete Distribution

Suppose the CDF of a discrete random variable X is given as:

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

Following the steps outlined above:

1. Discrete Values: The discrete values are 0, 1, 2, and 3.

2. Calculate Probabilities:

   • P(X=0) = F(0) - F(-∞) = 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

3. Verify Sum: 0.2 + 0.4 + 0.3 + 0.1 = 1. The sum is correct.

Therefore, the PMF is: P(X=0) = 0.2, P(X=1) = 0.4, P(X=2) = 0.3, P(X=3) = 0.1

Example 2: CDF with Gaps

Consider a CDF with gaps:

• F(1) = 0.3
• F(3) = 0.7
• F(5) = 1

Notice that there are no values defined for x=2 or x=4. In such a case, we assume the probability at those points is 0.

1. Discrete Values: The discrete values are 1, 3, and 5.

2. Calculate Probabilities:

   • P(X=1) = F(1) - F(-∞) = 0.3 - 0 = 0.3
   • P(X=3) = F(3) - F(1) = 0.7 - 0.3 = 0.4
   • P(X=5) = F(5) - F(3) = 1 - 0.7 = 0.3

   P(X=2) = P(X=4) = 0

3. Verify Sum: 0.3 + 0 + 0.4 + 0 + 0.3 = 1. The sum is correct.

Therefore, the PMF is: P(X=1) = 0.3, P(X=3) = 0.4, P(X=5) = 0.3, P(X=2) = P(X=4) = 0. Notice how the gaps in the CDF translate into zero probabilities in the PMF.

Example 3: More Complex Scenario

Let's consider a more complex CDF defined piecewise:

• F(x) = 0, for x < 1
• F(x) = 0.2x - 0.2, for 1 ≤ x < 2
• F(x) = 0.8, for 2 ≤ x < 3
• F(x) = x - 2.2, for 3 ≤ x < 4
• F(x) = 1, for x ≥ 4

This requires careful evaluation at the boundaries. Let's find the PMF for the integers within the range:

1. Discrete Values: The discrete values are 1, 2, 3, and 4.

2. Calculate Probabilities:

• P(X=1) = F(1) - F(0) = (0.2*1 - 0.2) - 0 = 0
• P(X=2) = F(2) - F(1) = 0.8 - (0.2*1 - 0.2) = 0.8
• P(X=3) = F(3) - F(2) = (3-2.2) - 0.8 = 0
• P(X=4) = F(4) - F(3) = 1 - (4-2.2) = 0

Notice the seemingly paradoxical result of P(X=1) and P(X=3) being zero despite having values in the CDF function. This is because for the range 1 ≤ x < 2, the cumulative probability increases continuously, resulting in the probability mass concentrated at integer values other than 1 and 3 within this range. This illustrates that the difference in the CDF must be calculated specifically between consecutive integer values, and that a continuous CDF does not necessarily translate directly into a PMF with non-zero probabilities at each integer.

3. Verify Sum: 0 + 0.8 + 0 + 0 = 0.8. The sum is not 1! This highlights a critical point: the provided CDF is not valid because it doesn't properly represent a probability distribution. A proper CDF for a discrete variable should yield a sum of probabilities equal to 1 when converted to a PMF.

This example demonstrates the importance of verifying the validity of the CDF before attempting to derive the PMF. An incorrectly defined CDF will lead to an incorrect and unusable PMF.

Handling Continuous CDFs

It's crucial to remember that the method described above is specifically for discrete random variables. If you are dealing with a continuous random variable, you cannot directly obtain a PMF. Instead, you'll need to find the Probability Density Function (PDF), which is the derivative of the CDF. The PDF describes the probability density at a given point, not the probability of the variable taking on that exact value (which is 0 for continuous variables).

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 in this guide, you can efficiently convert the CDF into the PMF. Remember to always verify the validity of the CDF and ensure that the sum of probabilities in the derived PMF equals 1. This guide has provided examples ranging from simple to more complex scenarios, preparing you to handle various cases you might encounter in your studies or applications. Always be mindful of whether you are working with discrete or continuous variables, and apply the appropriate method for obtaining the probability distribution.