How To Find Pmf From Cdf

Probability mass function (PMF) and cumulative distribution function (CDF) are two fundamental concepts in probability theory and statistics. Understanding their relationship and how to derive one from the other is crucial for analyzing and interpreting data. While the CDF represents the probability that a random variable takes on a value less than or equal to a given point, the PMF specifically focuses on the probability that a discrete random variable takes on a particular value. This article provides a comprehensive guide on how to find the PMF from the CDF, covering various methods, explanations, and examples to ensure a thorough understanding.

Understanding PMF and CDF

Before diving into the methods for finding the PMF from the CDF, it is essential to have a clear understanding of what each function represents.

Probability Mass Function (PMF)

The Probability Mass Function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. In other words, if X is a discrete random variable, the PMF is defined as:

P(X = x)

Where:

• X is the discrete random variable.
• x is a specific value that the random variable can take.
• P(X = x) is the probability that X is equal to x.

The PMF has two essential properties:

1. The probability for each value must be between 0 and 1, inclusive.
2. The sum of the probabilities for all possible values must equal 1.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF), also known as the distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Mathematically, the CDF is defined as:

F(x) = P(X ≤ x)

Where:

• X is the random variable (either discrete or continuous).
• x is a specific value.
• F(x) is the probability that X is less than or equal to x.

The CDF has the following properties:

1. It is a non-decreasing function.
2. It ranges from 0 to 1.
3. F(-∞) = 0 and F(∞) = 1

Methods to Find PMF from CDF

When dealing with a discrete random variable, the PMF can be derived from the CDF using the following methods:

Method 1: Using Differences in CDF Values

The primary method to find the PMF from the CDF involves calculating the differences in the CDF values at each possible value of the discrete random variable.

Given the CDF F(x), the PMF P(X = x) can be found as follows:

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

Where:

• F(x) is the value of the CDF at x.
• F(x - 1) is the value of the CDF at the value immediately preceding x.

Steps to Apply This Method:

1. Identify the Possible Values: Determine all possible values that the discrete random variable X can take.
2. Obtain the CDF Values: Find the values of the CDF at each of these possible values.
3. Calculate the Differences: Compute the difference between the CDF value at each point and the CDF value at the point immediately before it. This difference gives the probability that X is equal to that specific value.

Example:

Suppose we have a discrete random variable X with possible values {1, 2, 3, 4}, and the CDF is given as:

• F(1) = 0.2
• F(2) = 0.5
• F(3) = 0.8
• F(4) = 1.0

To find the PMF:

• P(X = 1) = F(1) - F(0) = 0.2 - 0 = 0.2 (Assuming F(0) = 0)
• P(X = 2) = F(2) - F(1) = 0.5 - 0.2 = 0.3
• P(X = 3) = F(3) - F(2) = 0.8 - 0.5 = 0.3
• P(X = 4) = F(4) - F(3) = 1.0 - 0.8 = 0.2

Therefore, the PMF is:

• P(X = 1) = 0.2
• P(X = 2) = 0.3
• P(X = 3) = 0.3
• P(X = 4) = 0.2

Method 2: Using the Definition of CDF for Discrete Variables

The CDF for a discrete random variable is a step function. Each step occurs at a value that the random variable can take, and the size of the step is equal to the probability of that value.

Given the CDF F(x), the PMF P(X = x) can be defined as the size of the jump in the CDF at x:

P(X = x) = jump in F(x) at x

Steps to Apply This Method:

1. Examine the CDF: Look at the CDF to identify the points where it jumps.
2. Determine the Jump Sizes: Calculate the size of each jump. The jump size is the difference between the CDF value immediately after the jump and the CDF value immediately before the jump.
3. Assign Probabilities: Assign the jump size as the probability for the corresponding value of the random variable.

Example:

Suppose we have the following CDF for a discrete random variable X:

• F(x) = 0, for x < 1
• F(x) = 0.3, for 1 ≤ x < 3
• F(x) = 0.7, for 3 ≤ x < 5
• F(x) = 1.0, for x ≥ 5

To find the PMF:

• At x = 1, the jump size is 0.3 - 0 = 0.3, so P(X = 1) = 0.3
• At x = 3, the jump size is 0.7 - 0.3 = 0.4, so P(X = 3) = 0.4
• At x = 5, the jump size is 1.0 - 0.7 = 0.3, so P(X = 5) = 0.3

Therefore, the PMF is:

• P(X = 1) = 0.3
• P(X = 3) = 0.4
• P(X = 5) = 0.3

Method 3: Using the Right-Continuity Property of CDF

The CDF is right-continuous, which means that the limit of the CDF as x approaches a value a from the right is equal to the CDF's value at a. Mathematically:

lim (x→a⁺) F(x) = F(a)

The jump at a point a can be calculated as:

P(X = a) = F(a) - lim (x→a⁻) F(x)

Where:

• F(a) is the value of the CDF at a.
• lim (x→a⁻) F(x) is the limit of the CDF as x approaches a from the left.

Steps to Apply This Method:

1. Identify Points of Discontinuity: Determine the points at which the CDF is discontinuous. These are the values that the discrete random variable can take.
2. Calculate Left-Hand Limits: For each point of discontinuity a, find the limit of the CDF as x approaches a from the left.
3. Calculate Jump Sizes: Compute the difference between the value of the CDF at a and the left-hand limit at a. This difference is the probability that X is equal to a.

Example:

Consider the following CDF:

• F(x) = 0, for x < 2
• F(x) = 0.4, for 2 ≤ x < 4
• F(x) = 0.9, for 4 ≤ x < 6
• F(x) = 1.0, for x ≥ 6

To find the PMF:

• At x = 2:
  • F(2) = 0.4
  • lim (x→2⁻) F(x) = 0
  • P(X = 2) = F(2) - lim (x→2⁻) F(x) = 0.4 - 0 = 0.4
• At x = 4:
  • F(4) = 0.9
  • lim (x→4⁻) F(x) = 0.4
  • P(X = 4) = F(4) - lim (x→4⁻) F(x) = 0.9 - 0.4 = 0.5
• At x = 6:
  • F(6) = 1.0
  • lim (x→6⁻) F(x) = 0.9
  • P(X = 6) = F(6) - lim (x→6⁻) F(x) = 1.0 - 0.9 = 0.1

Therefore, the PMF is:

• P(X = 2) = 0.4
• P(X = 4) = 0.5
• P(X = 6) = 0.1

Practical Examples

To further illustrate these methods, let’s consider a few practical examples:

Example 1: Rolling a Fair Six-Sided Die

Suppose we roll a fair six-sided die. The possible outcomes are {1, 2, 3, 4, 5, 6}, each with a probability of 1/6. The CDF can be constructed as follows:

• F(x) = 0, for x < 1
• F(x) = 1/6, for 1 ≤ x < 2
• F(x) = 2/6, for 2 ≤ x < 3
• F(x) = 3/6, for 3 ≤ x < 4
• F(x) = 4/6, for 4 ≤ x < 5
• F(x) = 5/6, for 5 ≤ x < 6
• F(x) = 1, for x ≥ 6

Using Method 2 (Jump Sizes):

• P(X = 1) = 1/6 - 0 = 1/6
• P(X = 2) = 2/6 - 1/6 = 1/6
• P(X = 3) = 3/6 - 2/6 = 1/6
• P(X = 4) = 4/6 - 3/6 = 1/6
• P(X = 5) = 5/6 - 4/6 = 1/6
• P(X = 6) = 1 - 5/6 = 1/6

The PMF is:

• P(X = 1) = 1/6
• P(X = 2) = 1/6
• P(X = 3) = 1/6
• P(X = 4) = 1/6
• P(X = 5) = 1/6
• P(X = 6) = 1/6

Example 2: Number of Heads in Two Coin Flips

Consider flipping a fair coin twice. Let X be the number of heads. The possible values for X are {0, 1, 2}. The probabilities are:

• P(X = 0) = 1/4 (TT)
• P(X = 1) = 2/4 = 1/2 (HT, TH)
• P(X = 2) = 1/4 (HH)

The CDF can be constructed as follows:

• F(x) = 0, for x < 0
• F(x) = 1/4, for 0 ≤ x < 1
• F(x) = 3/4, for 1 ≤ x < 2
• F(x) = 1, for x ≥ 2

Using Method 1 (Differences in CDF Values):

• P(X = 0) = F(0) - F(-1) = 1/4 - 0 = 1/4
• P(X = 1) = F(1) - F(0) = 3/4 - 1/4 = 1/2
• P(X = 2) = F(2) - F(1) = 1 - 3/4 = 1/4

The PMF is:

• P(X = 0) = 1/4
• P(X = 1) = 1/2
• P(X = 2) = 1/4

Example 3: A Discrete Random Variable with Given CDF

Suppose we have the following CDF for a discrete random variable X:

• F(x) = 0, for x < -1
• F(x) = 0.2, for -1 ≤ x < 0
• F(x) = 0.5, for 0 ≤ x < 1
• F(x) = 0.8, for 1 ≤ x < 2
• F(x) = 1, for x ≥ 2

Using Method 3 (Right-Continuity Property):

• At x = -1:
  • F(-1) = 0.2
  • lim (x→-1⁻) F(x) = 0
  • P(X = -1) = 0.2 - 0 = 0.2
• At x = 0:
  • F(0) = 0.5
  • lim (x→0⁻) F(x) = 0.2
  • P(X = 0) = 0.5 - 0.2 = 0.3
• At x = 1:
  • F(1) = 0.8
  • lim (x→1⁻) F(x) = 0.5
  • P(X = 1) = 0.8 - 0.5 = 0.3
• At x = 2:
  • F(2) = 1
  • lim (x→2⁻) F(x) = 0.8
  • P(X = 2) = 1 - 0.8 = 0.2

The PMF is:

• P(X = -1) = 0.2
• P(X = 0) = 0.3
• P(X = 1) = 0.3
• P(X = 2) = 0.2

Common Pitfalls and How to Avoid Them

When finding the PMF from the CDF, several common pitfalls can lead to incorrect results. Being aware of these pitfalls and knowing how to avoid them is essential for accurate analysis.

1. Incorrectly Identifying Jump Points:

   • Pitfall: Missing or misidentifying the points at which the CDF jumps.
   • Solution: Carefully examine the CDF to identify all points of discontinuity. Ensure that you account for every jump.

2. Miscalculating Jump Sizes:

   • Pitfall: Incorrectly calculating the size of the jumps, especially when the CDF is complex.
   • Solution: Double-check the CDF values immediately before and after each jump. Use the correct subtraction order (F(x) - F(x-1)).

3. Forgetting the Initial Value:

   • Pitfall: Failing to account for the initial value of the CDF, which is often 0.
   • Solution: Always remember that F(-∞) = 0. The first value in the PMF is the CDF value at the smallest possible value of the random variable.

4. Assuming Continuity:

   • Pitfall: Applying methods suitable for continuous random variables to discrete random variables.
   • Solution: Remember that the PMF is only defined for discrete random variables. The methods for finding PDFs from CDFs for continuous variables are different (differentiation).

5. Not Verifying Probabilities:

   • Pitfall: Failing to verify that the probabilities in the PMF sum up to 1.
   • Solution: Always check that Σ P(X = x) = 1. If the sum is not equal to 1, there is an error in the calculations.

Advanced Considerations

Dealing with Complex CDFs

In some cases, the CDF may be more complex, involving multiple steps or piecewise functions. In such situations, it is essential to proceed systematically.

1. Break Down the CDF: Divide the CDF into simpler segments based on the different ranges of values.
2. Apply Appropriate Methods: Apply the methods described above to each segment, ensuring that you correctly identify the jump points and calculate the jump sizes.
3. Combine Results: Combine the results from each segment to obtain the complete PMF.

Using Software and Tools

Various software and tools can assist in finding the PMF from the CDF, especially for complex distributions.

• Statistical Software: Programs like R, Python (with libraries such as NumPy, SciPy, and Matplotlib), and MATLAB can be used to plot the CDF and calculate the PMF.
• Spreadsheet Software: Programs like Microsoft Excel or Google Sheets can be used for simpler CDFs.

Example using Python:

import numpy as np
import matplotlib.pyplot as plt

# Given CDF values
x_values = np.array([1, 2, 3, 4])
cdf_values = np.array([0.2, 0.5, 0.8, 1.0])

# Calculate PMF
pmf_values = np.diff(cdf_values, prepend=0)

# Print PMF values
print("PMF Values:", pmf_values)

# Plot PMF
plt.bar(x_values, pmf_values)
plt.xlabel("X")
plt.ylabel("P(X = x)")
plt.title("Probability Mass Function")
plt.xticks(x_values)
plt.show()

Conclusion

Finding the PMF from the CDF is a fundamental skill in probability and statistics. By understanding the definitions of PMF and CDF and applying the methods described in this article, you can accurately derive the PMF from the CDF for discrete random variables. Whether using differences in CDF values, jump sizes, or the right-continuity property, it is essential to proceed systematically and double-check your calculations. Being aware of common pitfalls and using software tools when necessary will further enhance your ability to work with these essential statistical functions. The PMF provides a clear and concise representation of the probabilities associated with each value of a discrete random variable, making it an invaluable tool for data analysis and decision-making.