Marginal and Conditional Distributions
We will cover following topics
Introduction
In this chapter, we will delve into the concepts of marginal and conditional distributions for discrete bivariate random variables. Bivariate random variables involve two variables that are dependent on each other, and understanding their individual distributions and their relationships is crucial for various applications in statistics and probability.
When dealing with bivariate random variables, it’s important to consider both the individual behavior of each variable and how they interact together. Marginal and conditional distributions provide us with insights into these aspects.
Marginal Distribution
The marginal distribution of a single variable within a bivariate distribution is simply its distribution on its own, ignoring the other variable. In other words, we’re looking at the probability distribution of one variable while disregarding the other.
For instance, if we have a bivariate distribution of X and Y, the marginal distribution of $X$ can be obtained by summing up the probabilities of all the outcomes for different values of X while keeping $Y$ constant. This gives us the probability mass function of $X$.
Conditional Distribution
The conditional distribution focuses on the distribution of one variable given the value of the other variable. It tells us how one variable behaves when the other is fixed. Mathematically, the conditional distribution of $X$ given $Y=y$ is written as $P(X=x \mid Y=y)$.
Calculating the conditional distribution involves using the joint probability distribution and dividing it by the sum of probabilities for that specific value of $Y$. This gives us the probability of $X$ taking a particular value given that $Y$ has a fixed value.
Example: Let’s consider an example where X represents the number of heads obtained in two coin tosses, and Y represents the number of tails. We have the following joint probability distribution:
| X\Y | 0 | 1 | 2 |
|---|---|---|---|
| 0 | $\frac{1}{4}$ | $\frac{1}{4}$ | 0 |
| 1 | $\frac{1}{4}$ | $\frac{1}{4}$ | $\frac{1}{4}$ |
| 2 | 0 | $\frac{1}{4}$ | $\frac{1}{4}$ |
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The marginal distribution of $X$ is obtained by summing the probabilities in each column. For instance, $P(X=0)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}.$
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The conditional distribution of $X$ given $Y=1$ is $P(X=x \mid Y=1)$. So, $P(X=0 \mid Y=1)= \frac{1}{4}$, and $P(X=1 \mid Y=1)=\frac{1}{4}.$
Conclusion
Understanding the marginal and conditional distributions of bivariate random variables provides us with insights into the individual behavior of variables and their interactions. These distributions play a crucial role in various statistical analyses and decision-making processes. By calculating and interpreting these distributions, we can gain a deeper understanding of the underlying probabilistic relationships between variables.