Mean ,Variance & CDF of Discrete Random Variable| MAT202| MAT204 |MAT208 | MAT212 | Module 1| Part 2

Introduction

This tutorial will guide you through understanding the mean, variance, and cumulative distribution function (CDF) of discrete random variables. These concepts are fundamental in probability and statistics, especially for courses like MAT202, MAT204, MAT208, and MAT212. By the end of this tutorial, you will have a clearer understanding of how to calculate and interpret these statistical measures.

Step 1: Understanding Discrete Random Variables

• A discrete random variable is a variable that can take on a countable number of distinct values.
• Examples include the number of heads in a series of coin flips or the number of students in a classroom.
• Familiarize yourself with the concept of a probability mass function (PMF), which assigns probabilities to each possible value of the random variable.

Step 2: Calculating the Mean

• The mean (or expected value) of a discrete random variable is calculated using the formula:

  [ E(X) = \sum_{i=1}^{n} x_i \cdot P(X = x_i) ]

  where ( x_i ) are the values of the random variable and ( P(X = x_i) ) is the probability of each value.

• Steps to calculate:

  1. List all possible values of the random variable.
  2. Determine the probability for each value.
  3. Multiply each value by its corresponding probability.
  4. Sum all the products to find the mean.

Step 3: Calculating the Variance

• Variance measures the spread of the random variable's values around the mean and is given by:

  [ Var(X) = E(X^2) - [E(X)]^2 ]

• Steps to calculate:

  1. Calculate the mean as described in Step 2.
  2. Compute ( E(X^2) ) using the formula:

  [ E(X^2) = \sum_{i=1}^{n} x_i^2 \cdot P(X = x_i) ]

  3. Subtract the square of the mean from ( E(X^2) ) to get the variance.

Step 4: Understanding the Cumulative Distribution Function

• The CDF of a discrete random variable is defined as:

  [ F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t) ]

• Steps to calculate the CDF:

  1. For each value of the random variable, sum the probabilities of all outcomes less than or equal to that value.
  2. Create a table to visualize the CDF, showing values of ( x ) and their corresponding ( F(x) ).

Step 5: Exploring Common Discrete Distributions

• Binomial Distribution: Used for a fixed number of independent trials, each with the same probability of success.

• Poisson Distribution: Used to model the number of events occurring in a fixed interval of time or space.

• Understand the parameters of these distributions:

  • Binomial: number of trials (n) and probability of success (p).
  • Poisson: average rate (λ) of occurrence.

Conclusion

In this tutorial, you learned about discrete random variables, how to calculate their mean and variance, and how to construct a cumulative distribution function. These concepts are crucial for analyzing data in various fields. To further your understanding, consider applying these calculations to real-world data sets, or explore additional examples of binomial and Poisson distributions to solidify your knowledge.