Cumulative Distribution Function

Cumulative Distribution Function (CDF), is a fundamental concept in probability theory and statistics that provides a way to describe the distribution of the random variable. It represents the probability that a random variable takes a value less than or equal to a certain value. The CDF is a non-decreasing function that ranges from 0 to 1 capturing the entire probability distribution of the random variable.

What is a Cumulative Distribution Function?

The Cumulative Distribution Function (CDF) of a random variable is a mathematical function that provides the probability that the variable will take a value less than or equal to a particular number.

The CDF starts at 0 for the smallest possible value of X and increases to 1 as x approaches the largest possible value of X. It is a non-decreasing function that provides a complete description of the distribution of the random variable.

For example, if you're looking at the CDF for a test score of 80, and it gives you 0.75, this means there's a 75% chance that a random student's score will be 80 or less.

In short, the CDF helps you understand the likelihood of a random value being within a certain range by summing up probabilities as you go along.

Definition of CDF

The Cumulative Distribution Function F(x) of the random variable X is defined as:

F(x) = P(X \leq x)

Graphical Representation of Cumulative Distribution Function

The graph of a CDF is a non-decreasing curve that starts at 0 and approaches 1 as the x increases. It is useful for visualizing the probability distribution of the random variable.

Properties of Cumulative Distribution Function

Some of the properties of Cumulative Distribution Function:

• Monotonicity

The CDF is a non-decreasing function. This means that for any two values x_1 and x_2 such that x_1 \leq x_2 the corresponding CDF values satisfy F(x_1) \leq F(x_2).

• Limits

As x approaches negative infinity the CDF approaches 0:

\lim_{x \to -\infty} F(x) = 0

As x approaches positive infinity the CDF approaches 1:

\lim_{x \to +\infty} F(x) = 1

• Continuity

The CDF of a continuous random variable is a continuous function while the CDF of the discrete random variable has jumps or discontinuities at specific points.

• Non-Negativity

The CDF is always non-negative i.e. F(x) \geq 0 for the all x.

How to Calculate Cumulative Distribution Function?

Steps to find cumulative distribution function are given below-

Step 1. Identify the Distribution: The Determine whether the random variable follows a discrete or continuous distribution.

Step 2. Determine the PDF: For continuous distributions find the Probability Density Function (PDF) f(x).

Step 3. Integrate the PDF: The Integrate the PDF from the -\infty to x to find the CDF:

F(x) = \int_{-\infty}^{x} f(t) \, dt

Step 4. Sum the Probabilities: For discrete distributions sum the probabilities for the all values less than or equal to the x.

Using Probability Density Function (PDF) to Find CDF

For continuous random variables the CDF F(x) is derived from the PDF f(x) by the integrating:

F(x) = \int_{-\infty}^{x} f(t) \, dt

This process accumulates the probability from the left up to the point x.

Types of Cumulative Distribution Functions

CDF of a Discrete Random Variable

For a discrete random variable X with the possible values x_1, x_2, \dots, x_n and corresponding probabilities P(X = x_i) = p_i the CDF is given by:

F(x) = \sum_{x_i \leq x} p_i

The CDF of the discrete random variable increases in steps at the points where the variable takes on the specific values.

For example, Consider a random variable X that represents the roll of the fair 6-sided die. The CDF F(x) for this random variable is:

F(x) = \begin{cases} 0 & \text{if } x < 1 \\ \frac{1}{6} & \text{if } 1 \leq x < 2 \\ \frac{2}{6} & \text{if } 2 \leq x < 3 \\ \frac{3}{6} & \text{if } 3 \leq x < 4 \\ \frac{4}{6} & \text{if } 4 \leq x < 5 \\ \frac{5}{6} & \text{if } 5 \leq x < 6 \\ 1 & \text{if } x \geq 6 \end{cases}

CDF of a Continuous Random Variable

For a continuous random variable X with the probability density function (PDF) f(x) the CDF is the integral of the PDF:

F(x) = \int_{-\infty}^{x} f(t)\, dt

Since the CDF is the integral of the PDF it is a continuous and differentiable function.

For example: Consider a continuous random variable X that is uniformly distributed between the 0 and 1. The CDF F(x) for this random variable is:

F(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x \leq 1 \\ 1 & \text{if } x > 1 \end{cases}

Difference Between Cumulative Distribution Function (CDF) and Probability Density Function (PDF)

Some of the common differences between CDF and PDF are listed in the following table:

When to Use CDF vs. PDF?

• CDF: Use the CDF when we need to find the probability that a random variable is less than or equal to the specific value. It provides the cumulative probability up to the certain point.
• PDF: Use the PDF when you need to the find the probability density at a specific value. The PDF represents the rate of the change of the CDF and is useful for the calculating the probabilities over intervals.

Relationship Between CDF and PDF

For a continuous random variable X with the PDF f(x) the relationship between the CDF F(x) and PDF is given by:

f(x) = \frac{dF(x)}{dx}

This indicates that the PDF is the derivative of the CDF.

Solved Examples: Cumulative Distribution Function

Example 1: Consider a random variable X representing the number of the heads obtained in the two coin flips. Calculate the CDF F(x) of X.

Solution:

The possible values of X are 0, 1 and 2. The probabilities are:

P(X=0) = \frac{1}{4} \quad \text{(no heads)}

P(X=1) = \frac{1}{2} \quad \text{(one head)}

P(X=2) = \frac{1}{4} \quad \text{(two heads)}

The CDF F(x) is:

F(x) = \begin{cases} 0 & \text{if } x < 0 \\ \frac{1}{4} & \text{if } 0 \leq x < 1 \\ \frac{3}{4} & \text{if } 1 \leq x < 2 \\ 1 & \text{if } x \geq 2 \\ \end{cases}

Example 2: Let X be a continuous random variable with the PDF f(x) = 2x for the 0 \leq x \leq 1. Find the CDF F(x).

Solution:

To find the CDF integrate the PDF:

F(x) = \int_{-\infty}^{x} f(t) \, dt = \int_{0}^{x} 2t \, dt = x^2 \quad \text{for } 0 \leq x \leq 1

Thus, the CDF is:

F(x) = \begin{cases} 0 & \text{if } x < 0 \\ x^2 & \text{if } 0 \leq x \leq 1 \\ 1 & \text{if } x > 1 \\ \end{cases}

Example 3: Given the CDF F(x) = 1 - e^{-x} for x \geq 0 find the probability that X lies between the 1 and 2.

Solution:

The probability that X lies between the 1 and 2 is:

P(1 \leq X \leq 2) = F(2) - F(1)

Calculate:

F(2) = 1 - e^{-2} \quad \text{and} \quad F(1) = 1 - e^{-1}

P(1 \leq X \leq 2) = (1 - e^{-2}) - (1 - e^{-1}) = e^{-1} - e^{-2}

Example 4: Let X be a mixed random variable with the following distribution: P(X=0) = 0.2, P(X=1) = 0.3 and the continuous part is uniformly distributed over [2, 3]. Find the CDF F(x).

Solution:

For x < 0, F(x) = 0.

For 0 \leq x < 1, F(x) = 0.2.

For 1 \leq x < 2, F(x) = 0.5.

For 2 \leq x < 3 the continuous part applies:

F(x) = 0.5 + 0.5 \times (x - 2) = 0.5 + 0.5x - 1

Thus, F(x) = 0.5x - 0.5 for 2 \leq x \leq 3.

For x \geq 3, F(x) = 1.

Example 5: Given a PDF f(x) = \frac{3}{4}(1 - x^2) for the -1 \leq x \leq 1 find the CDF F(x).

Solution:

Integrate the PDF to find the CDF:

F(x) = \int_{-\infty}^{x} f(t) \, dt = \int_{-1}^{x} \frac{3}{4}(1 - t^2) \, dt

Solving the integral:

F(x) = \frac{3}{4} \left[t - \frac{t^3}{3}\right]_{-1}^{x} = \frac{3}{4} \left(x - \frac{x^3}{3} + \frac{2}{3}\right)

The CDF (F(x)) is:

F(x) = \begin{cases} 0 & \text{if } x < -1 \\ \frac{3}{4} \left(x - \frac{x^3}{3} + \frac{2}{3}\right) & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } x > 1 \\ \end{cases}

Practical Question: Cumulative Distribution Function (CDF)

Questions 1. Find the CDF of a random variable X representing the number of the heads in three coin flips.

Questions 2. Determine the CDF for a continuous random variable X with the PDF f(x) = 3x^2 for 0 \leq x \leq 1.

Questions 3. Calculate the probability that X is less than or equal to the 4 given the CDF F(x) = 1 - e^{-x} for x \geq 0.

Questions 4. Find the CDF for the random variable X uniformly distributed over the interval [2, 5].

Questions 5. Given the CDF F(x) for a random variable X find the PDF by differentiating the CDF.

Questions 6. Use the CDF F(x) = 1 - e^{-x} to find the probability that X lies between the 2 and 3.

Questions 7. For a discrete random variable X taking values 1, 2 and 3 with the probabilities 0.2, 0.5 and 0.3 respectively find the CDF F(x).

Questions 8.Calculate the CDF for a normal distribution with the mean 0 and standard deviation 1 at x = 1.

Questions 9.Given a random variable X with the PDF f(x) = \frac{1}{2} for 0 \leq x \leq 2 find the CDF F(x).

Questions 10. Determine the probability that a continuous random variable X with the PDF f(x) = 4x for the 0 \leq x \leq 1 is greater than 0.5.

Conclusion

The Cumulative Distribution Function (CDF) is an essential concept in the probability and statistics offering the comprehensive way to the describe the distribution of the random variables. Understanding the CDF and its properties is fundamental to the various statistical analyses and real-world applications. Whether dealing with the discrete or continuous variables the CDF provides the powerful tool for the calculating probabilities and making the inferences about data.

Read More

• Probability Distribution
• Probability Distribution Function
• What is the difference between CDF vs PDF
• Difference between Average and Mean
• Difference between Variance and Standard deviation