14.8. Example Problems

Given:

1 𝑃(𝑋 = 𝑥,𝑌 = 𝑦) = for (𝑥,𝑦) ∈ {(0,0),(0,1),(1,0),(1,1)} 8

• Find marginal distributions • Check independence

Solution:

1 1 1 • 𝑃 (0)= 𝑃(0,0)+𝑃(0,1)= + = 𝑋 8 8 4

1 1 1 • 𝑃 (0)= 𝑃(0,0)+ 𝑃(1,0) = + = 𝑌 8 8 4

1 1 1 1 • Since 𝑃(0,0) = ≠ 𝑃 (0)⋅𝑃 (0) = ⋅ = , not independent. 𝑋 𝑌 8 4 4 16

In this problem, we are given a joint probability mass function (pmf) for two discrete random variables, X and Y. The probabilities are non-negative for each combination of (x, y). We first calculate the marginal distributions for X and Y by summing the probabilities over all values of the other variable. To check for independence, we evaluate whether the joint probability equals the product of the marginals; if they are not equal, X and Y are dependent.

Think of X and Y as two different types of dice—X could represent the outcome of a die that only shows even numbers (0,2), while Y shows odd numbers (1,3). The joint pmf describes the likelihood of rolling specific pairs of these dice. If you look at just one die (get the marginal distribution), it could just be viewed on its own. However, the relationship between the two shows whether the dice influence each other or if they roll independently.

Let:

𝑓(𝑥,𝑦) = 4𝑥𝑦 for 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1

• Check validity • Find marginal distributions

Solution:

• Check total probability:

1 1 ∬𝑓(𝑥,𝑦) 𝑑𝑥 𝑑𝑦 = ∫ ∫ 4𝑥𝑦 𝑑𝑥 𝑑𝑦 = 1 ✓ 0 0

• Marginal of 𝑋:

1 1 𝑓 (𝑥) = ∫ 4𝑥𝑦 𝑑𝑦 = 4𝑥⋅ = 2𝑥 𝑋 2 0

• Marginal of 𝑌:

1 1 𝑓 (𝑦) = ∫ 4𝑥𝑦 𝑑𝑥 = 4𝑦 ⋅ = 2𝑦 𝑌 2 0

• Since 𝑓(𝑥,𝑦) = 𝑓 (𝑥)⋅𝑓 (𝑦), 𝑋 and 𝑌 are independent. 𝑋 𝑌

In this continuous case example, we start with a joint probability density function (pdf). To ensure that it forms a valid probability distribution, we integrate the joint pdf over the entire range and confirm that the total probability equals 1. Next, we find the marginal distributions for X and Y by integrating the joint pdf with respect to the other variable. Finally, we check for independence by confirming whether the joint pdf can be expressed as a product of the marginal pdfs.

Imagine you have a garden where x represents the amount of sunlight and y represents the amount of water each plant receives. The joint pdf represents how these two factors interact to affect plant growth. By assessing the total influence and then breaking down how much each individual factor contributes, we can understand if one factor can be changed independently of the other or whether adjusting one inevitably affects the other.

• Joint Probability Distributions help us analyze the relationship between two or more random variables. • The joint PMF (for discrete) or joint PDF (for continuous) defines the likelihood of outcomes for pairs of random variables. • Marginal distributions give the distribution of a single variable irrespective of the other. • Conditional distributions help in understanding dependencies. • Independence implies the joint distribution is the product of the marginals.

The examples show how joint probability distributions can express relationships between variables, be they discrete or continuous. Marginal distributions reflect individual behaviors regardless of the relationship; conditional distributions indicate how one variable can impact another. Independence simplifies analysis, allowing separation of the randomness associated with each variable.

In a weather forecasting scenario, consider temperature and humidity. The joint distribution offers insights into various combinations of numbers, but we can separate them to consider only temperatures or humidities (marginals). Recognizing whether the temperature affects humidity (conditional) or if they are entirely independent simplifies our models and predictions significantly.

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