In Exercise 5.1 we determined that the joint distribution of...

In Exercise 5.1 we determined that the joint distribution of $Y_{1}$, the number of contracts awarded to firm $\mathrm{A},$ and $Y_{2},$ the number of contracts awarded to firm $\mathrm{B}$, is given by the entries in the following table. $$\begin{array}{cccc}\hline & \multicolumn{3}{c} {y_{1}} \\\)\cline { 2 - 4 }\( y_{2} & 0 & 1 & 2 \\\hline 0 & 1 / 9 & 2 / 9 & 1 / 9 \\1 & 2 / 9 & 2 / 9 & 0 \\2 & 1 / 9 & 0 & 0 \\\hline\end{array}$$ The marginal probability function of $Y_{1}$ was derived in Exercise 5.19 to be binomial with $n=2$ and $p=1 / 3 .$ Are $Y_{1}$ and $Y_{2}$ independent? Why?

In Exercise 5.1 we determined that the joint distribution of $Y_{1}$, the number of contracts awarded to firm $\mathrm{A},$ and $Y_{2},$ the number of contracts awarded to firm $\mathrm{B}$, is given by the entries in the following table.
$$\begin{array}{cccc}\hline & \multicolumn{3}{c} {y_{1}} \\\)\cline { 2 - 4 }\( y_{2} & 0 & 1 & 2 \\\hline 0 & 1 / 9 & 2 / 9 & 1 / 9 \\1 & 2 / 9 & 2 / 9 & 0 \\2 & 1 / 9 & 0 & 0 \\\hline\end{array}$$
The marginal probability function of $Y_{1}$ was derived in Exercise 5.19 to be binomial with $n=2$ and $p=1 / 3 .$ Are $Y_{1}$ and $Y_{2}$ independent? Why?

Key Concepts

Joint Distribution

A joint probability distribution describes the likelihood of different outcomes for two or more random variables considered at the same time. It provides a complete picture of how the variables interact, which is essential for analyzing the overall behavior of a system with multiple random factors.

Marginal Distribution

A marginal distribution is derived from a joint distribution and represents the probabilities of a single random variable irrespective of the values of the other variables. This is obtained by summing or integrating out the other variables, and it is useful for understanding the individual behavior of a variable within a multivariate setup.

Independence of Random Variables

Two random variables are independent if the joint probability of any combination of their outcomes equals the product of their marginal probabilities. This property indicates that the outcome of one variable does not influence the outcome of the other. Establishing whether variables are independent often involves testing if this multiplicative condition holds across all possible pairs of outcomes.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is commonly characterized by two parameters—the number of trials and the probability of success—and is widely used in scenarios where each trial has only two possible outcomes.

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