A Step-by-Step Guide to Random Variables: The Long Jump Tournament Tutorial

Photo by Paul Wong on Unsplash

Welcome to our intuitive guide to random variables, inspired by the story of the long jump tournament. In this tutorial, we will walk you through the steps to understand and analyze the relationship between two random variables, using the examples from the long jump tournament. Let’s dive in!

Step 1: Define Random Variables and Joint Probability Distribution

In the long jump tournament, we have two random variables:

1. ε — the participant’s height
2. n — the jump length

We will use a table of joint probability distribution to represent the probabilities of different combinations of heights and jump lengths. The table will look like this:

Step 2: Find the Marginal Probability Distribution

To analyze the individual characteristics of the two random variables, we need to find the marginal probability distributions for both ε and n. To do this, add up the probabilities across all jump lengths for each height and across all heights for each jump length.

For ε: P(ε = 164) = 0.03 + 0.11 = 0.14

P(ε = 176) = 0.08 + 0.16 = 0.24

P(ε = 190) = 0.14 + 0.08 = 0.22

P(ε = 198) = 0.14 + 0.26 = 0.40

For n: P(n = 4) = 0.03 + 0.08 + 0.14 + 0.14 = 0.39

P(n = 5) = 0.11 + 0.16 + 0.08 + 0.26 = 0.61

Step 3: Calculate Expected Values

The expected value represents the “average” value for each random variable. To calculate the expected value, multiply each value of the random variable by its corresponding probability and sum up the results.

For ε: E(ε) = (164 * 0.14) + (176 * 0.24) + (190 * 0.22) + (198 * 0.40) = 186.2

For n: E(n) = (4 * 0.39) + (5 * 0.61) = 4.61

Step 4: Calculate Variance

The variance measures the spread of the data. To find the variance of the random variables, calculate the expected value of the squared random variable (E(ε²) or E(n²)) and subtract the square of the expected value (E(ε)² or E(n)²).

For ε: Var(ε) = E(ε²) — (E(ε))² = 34823.28–34652.84 = 152.84

For n: Var(n) = E(n²) — (E(n))² = 21.49–21.2521 = 0.2379

Step 5: Calculate Covariance

To measure the degree to which the two random variables move together, we need to calculate the covariance. To do this, first find the expected value of their product E(εn), and then use the formula: Cov(ε, n) = E(εn) — E(ε)E(n).

Cov(ε, n) = E(εn) — E(ε)E(n) = 857.68 — (186.2 * 4.61) = 857.68–858.382 = -0.702

Step 6: Calculate the Correlation Coefficient

The correlation coefficient is a standardized measure of the relationship between the two random variables. To calculate the correlation coefficient ρ(ε, n), use the following formula:

ρ(ε, n) = Cov(ε, n) / (σ(ε) * σ(n))

where σ(ε) and σ(n) are the standard deviations of ε and n, respectively.

First, calculate the standard deviations of ε and n:

σ(ε) = sqrt(Var(ε)) = sqrt(152.84) ≈ 12.36 σ(n) = sqrt(Var(n)) = sqrt(0.2379) ≈ 0.488

Now, calculate the correlation coefficient ρ(ε, n):

ρ(ε, n) = -0.702 / (12.36 * 0.488) ρ(ε, n) = -0.702 / 6.032 ρ(ε, n) ≈ -0.116

Conclusion

In this tutorial, we explored the fascinating world of random variables using the example of a long jump tournament. By going through each step, we gained insights into the individual characteristics of the random variables (height and jump length), as well as their relationship.

We calculated the expected values, variance, covariance, and correlation coefficient to better understand the data. This tutorial demonstrated the power and usefulness of mathematical tools in analyzing real-world scenarios.

Now that you’ve learned how to work with random variables, you can apply these techniques to other situations in your life, whether it’s sports, finance, or any other field where you need to analyze the relationship between different variables. Happy analyzing, and thank you for your attention.