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

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. The marginal probability function of $Y_{1}$ was derived in Exercise 5.19 to be binomial with $n=2$ and $p=1 / 3 .$ Find a. $E\left(Y_{1}\right)$ b. $V\left(Y_{1}\right)$ c. $E\left(Y_{1}-Y_{2}\right)$

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. The marginal probability function of $Y_{1}$ was derived in Exercise 5.19 to be binomial with $n=2$ and $p=1 / 3 .$ Find a. $E\left(Y_{1}\right)$
b. $V\left(Y_{1}\right)$
c. $E\left(Y_{1}-Y_{2}\right)$

Key Concepts

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. Its key parameters are the number of trials (n) and the probability of success (p). This distribution is fundamental for modeling situations where outcomes are binary and trials are repeated in a consistent manner.

Expected Value

The expected value of a random variable is the long?term average outcome of a probability distribution. For a binomial random variable, the expected value is calculated as n multiplied by p. In any context, it serves as a measure of the central tendency or the mean of the distribution.

Variance

Variance is a measure of the spread or dispersion of a random variable's possible outcomes around its expected value. For a binomial random variable, the variance is derived from the formula n · p · (1-p), indicating how the outcomes fluctuate. Variance is essential for understanding the reliability and variability inherent in the process being modeled.

Linearity of Expectation

The linearity of expectation is a fundamental property that states the expected value of a sum (or difference) of random variables is equal to the sum (or difference) of their expected values, regardless of whether the variables are independent. This principle simplifies the computation of expectations in scenarios involving combined random variables.

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