Expected Value and Variance of Discrete Random Variables

Introduction

This tutorial introduces the concepts of expected value and variance for discrete random variables. Understanding these concepts is crucial for data analysis and statistics, as they provide insight into the behavior and characteristics of random variables. We will cover the formulas used to calculate expected value and variance, along with a practical example to illustrate these concepts.

Step 1: Understanding Expected Value

The expected value is a measure of the center of a probability distribution. It represents the long-term average outcome of a random variable.

Formula for Expected Value

The formula for calculating the expected value (E(X)) of a discrete random variable (X) is:

[ E(X) = \sum (x_i \cdot P(x_i)) ]

Where:

• (x_i) is a possible outcome.
• (P(x_i)) is the probability of that outcome.

Steps to Calculate Expected Value

1. List all possible outcomes of the random variable.
2. Determine the probability of each outcome.
3. Multiply each outcome by its corresponding probability.
4. Sum all the products to get the expected value.

Practical Advice

• Make sure the probabilities add up to 1.
• Double-check your calculations for accuracy.

Step 2: Understanding Variance

Variance measures the spread of a random variable's possible outcomes around the expected value. It indicates how much the values of a random variable differ from the expected value.

Formula for Variance

The variance (Var(X)) can be calculated using the formula:

[ Var(X) = \sum ((x_i - E(X))^2 \cdot P(x_i)) ]

Where:

• (E(X)) is the expected value calculated in Step 1.

Steps to Calculate Variance

1. Calculate the expected value using the steps outlined in Step 1.
2. For each outcome, subtract the expected value and square the result.
3. Multiply each squared result by the probability of that outcome.
4. Sum all these products to find the variance.

Practical Advice

• Variance is always non-negative.
• A high variance indicates a wide spread of outcomes, while a low variance indicates that the outcomes are closer to the expected value.

Step 3: Example Calculation

Let’s use a simple example to illustrate both concepts.

Given Data

• Possible outcomes: (x_1 = 1), (x_2 = 2), (x_3 = 3)
• Corresponding probabilities: (P(x_1) = 0.2), (P(x_2) = 0.5), (P(x_3) = 0.3)

Calculate Expected Value

1. (E(X) = (1 \cdot 0.2) + (2 \cdot 0.5) + (3 \cdot 0.3))
2. (E(X) = 0.2 + 1.0 + 0.9 = 2.1)

Calculate Variance

1. Calculate ((x_i - E(X))^2):
   • For (x_1): ((1 - 2.1)^2 = 1.21)
   • For (x_2): ((2 - 2.1)^2 = 0.01)
   • For (x_3): ((3 - 2.1)^2 = 0.81)
2. Calculate variance:
   • (Var(X) = (1.21 \cdot 0.2) + (0.01 \cdot 0.5) + (0.81 \cdot 0.3))
   • (Var(X) = 0.242 + 0.005 + 0.243 = 0.490)

Conclusion

In this tutorial, we covered the concepts of expected value and variance for discrete random variables, providing the formulas and a step-by-step example for each. Understanding these concepts enables you to analyze the behavior of random variables effectively. As a next step, practice calculating expected value and variance using different sets of data to reinforce your understanding.