91Ó°ÊÓ

The expected value, often symbolized as \(E(X)\), is a fundamental concept in probability and statistics. It is essentially the average or mean value that you would expect to see if you were to repeat an experiment or process a large number of times.
The formula for calculating the expected value of a discrete random variable \(X\) is given by:

• \(E(X) = \sum x \, p(x)\)

Here, \(x\) represents each possible value that the random variable can take, \(p(x)\) is the probability of \(x\) occurring, and the sum is over all possible values of \(x\).
In our example, where \(X\) is the number of 5-lb containers ordered, the expected value is calculated by summing the products of each \(x\) and its corresponding probability \(p(x)\):

• \(E(X) = 1(0.2) + 2(0.4) + 3(0.3) + 4(0.1) = 2.3\)

This tells us that, on average, a customer would order 2.3 containers.

Variance, denoted as \(V(X)\), measures the spread or variability of the random variable \(X\) around its expected value \(E(X)\). It's useful for understanding how much the values of \(X\) deviate from the expected value on average.
The formula for variance is:

• \(V(X) = E(X^2) - [E(X)]^2\)

To calculate it, you first need \(E(X^2)\), the expected value of the square of \(X\), which is:

• \(E(X^2) = \sum x^2 \, p(x)\)
• \( = 1^2(0.2) + 2^2(0.4) + 3^2(0.3) + 4^2(0.1) = 6.1\)

Using \(E(X^2)\) and \(E(X)\), we find the variance:

• \(V(X) = 6.1 - 2.3^2 = 0.81\)

This means that there is some variability in the number of containers that customers order, but not a lot.

The probability mass function (pmf) of a discrete random variable \(X\) provides the probabilities of each possible value \(X\) can take. It is a way to describe the distribution of a discrete random variable.
For a pmf, the following properties hold:

• \(0 \leq p(x) \leq 1\) for all \(x\)
• The sum of all probabilities must equal 1: \(\sum p(x) = 1\)

In the context of the chemical supply company's exercise, the pmf is:

• \(p(1) = 0.2\)
• \(p(2) = 0.4\)
• \(p(3) = 0.3\)
• \(p(4) = 0.1\)

This pmf shows the likelihood of a customer ordering 1, 2, 3, or 4 containers. Each probability indicates how often each order size is expected to occur in the long run. Understanding the pmf is key for deriving the expected value and variance, as it tells us the different values \(X\) can assume and with what frequency.