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The expected value in probability theory is a fundamental concept which gives us the long-term average or mean of a random variable. It serves as the center or the balancing point of the probability distribution. For any random variable, such as the number of lots ordered by a customer, the expected value provides a way of summarizing the distribution with a single number.

In our chemical supply problem, the random variable, denoted as \(X\), indicates how many 5-lb lots are ordered. To find the expected number of lots, we use the formula:

• \(E(X) = \sum x p(x)\), where \(x\) is the number of lots and \(p(x)\) is the probability of ordering that many lots.

Inserting the actual numbers, we get \(E(X) = 1 \times 0.2 + 2 \times 0.4 + 3 \times 0.3 + 4 \times 0.1 = 2.3\).

This means that, on average, a customer orders 2.3 lots of 5 lbs each from the chemical supply company.

Variance measures how much the values of a random variable differ from the expected value, or from each other. It gives us an idea about the spread or dispersion of the possible outcomes. The higher the variance, the more spread out the values are around the expected value.

The variance of a random variable \(X\), like the number of lots ordered, is computed using:

• \(V(X) = E(X^2) - [E(X)]^2\)

Here, \(E(X^2)\) is the expected value of the square of \(X\), calculated as \(E(X^2) = 1^2 \times 0.2 + 2^2 \times 0.4 + 3^2 \times 0.3 + 4^2 \times 0.1 = 6.1\).

So, \(V(X) = 6.1 - (2.3)^2 = 0.81\).

This means the number of lots ordered may slightly vary from the expected value of 2.3 with a variance of 0.81.

A random variable is a fundamental concept in probability, representing a variable that takes on different possible values, each with a probability. It provides a way to quantify and model variability and randomness.

In our chemical supply problem, \(X\) is the random variable that denotes the number of 5-lb lots ordered by a customer. This variable is described by a specific probability mass function (pmf), which lists the probabilities of \(X\) taking different values.

• The pmf for \(X\) is given as \[ \begin{array}{l|llll} x & 1 & 2 & 3 & 4 \ \hline p(x) & .2 & .4 & .3 & .1 \end{array} \] This indicates discrete outcomes, meaning there are limited specific values \(X\) can take, such as 1, 2, 3, or 4 lots, each with its probability.

Understanding these probabilities helps compute expected values, variances, and make decisions based on the likelihood of different outcomes.

The chemical supply problem is a practical application of probability and statistics in business, where a company assesses how much stock a customer might purchase. This involves analyzing their current inventory, which, in this case, is 100 lbs of chemicals sold in 5-lb lots, to predict future transactions and manage supply efficiently.

• The main objective is to determine the expected amount and variance of the chemical product left after meeting customer demand.

Initially, they calculate the expected number of lots ordered, \(E(X) = 2.3\). With each lot being 5 lbs, the expected remaining inventory after one customer's order is \(100 - 5E(X) = 88.5\) lbs.

To understand how much this expected value can vary from actual outcomes, they determine the variance in pounds left, calculated as \(5^2 \times V(X) = 20.25\). This predicts possible fluctuations in inventory, helping in making informed stock management decisions.