A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 8-lb batches. Let suppose that X has the following pmf = the number of batches ordered by a randomly chosen customer, and p(x) 0.3 0.5 0.1 01 Compute E(X) and VIX) E(X)2 их-0.8 batches batches Compute the expected number of pounds left after the next customer's order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of X.,] expected weight left variance of weight left lb lb

The expected weight left after the next customer's order shipped is 92 pounds, and the variance of the weight left is 51.2 lb²

What is the probability mass function?

A probability mass function (PMF) is a mathematical function that describes the probability of each possible value of a discrete random variable. In simpler terms, the PMF gives the likelihood of a specific outcome occurring when you randomly select an item from a finite set of possible outcomes.

The PMF assigns a probability to each outcome, such that the sum of all probabilities equals 1. The PMF is often presented in a table or a graph to show the probabilities associated with each possible value of the random variable.

The given probability mass function (pmf) for the number of batches ordered by a randomly chosen customer, X, is:

X P(X)

0 0.3

1 0.5

2 0.1

3 0.1

To calculate the expected value of X, we use the formula:

E(X) = Σ [x * P(X=x)]

E(X) = (0 * 0.3) + (1 * 0.5) + (2 * 0.1) + (3 * 0.1)

E(X) = 0 + 0.5 + 0.2 + 0.3

E(X) = 1

To calculate the variance of X, we first need to calculate the squared values of X:

X²

0² = 0

1²  = 1

2² = 4

3² = 9

Then, we use the formula:

Var(X) = E(X²) - [E(X)]²

E(X²) = Σ [x² * P(X=x)]

E(X²) = (0² * 0.3) + (1² * 0.5) + (2² * 0.1) + (3² * 0.1)

E(X²) = 0 + 0.5 + 0.4 + 0.9

E(X²) = 1.8

Var(X) = E(X²) - [E(X)]²

Var(X) = 1.8 - 1²

Var(X) = 0.8 batches²

The expected number of pounds left after the next customer's order is shipped can be calculated as:

E(Weight left) = 100 - 8X

where X is the number of batches ordered by the next customer.

Using the formula for the expected value of X that we found earlier, we can write:

E(Weight left) = 100 - 8E(X)

E(Weight left) = 100 - 8(1)

E(Weight left) = 92 lb

To calculate the variance of the number of pounds left, we first need to find the variance of X, which we already calculated as 0.8 batches² Then, we use the formula for the variance of a linear function of a random variable:

Var(aX + b) = a² * Var(X)

where a = -8 (the negative sign indicates that the weight left decreases as X increases) and b = 100.

Var(Weight left) = (-8)² * Var(X)

Var(Weight left) = 64 * 0.8

Var(Weight left) = 51.2 lb^2

Therefore, the expected weight left after the next customer's order is shipped is 92 pounds, and the variance of the weight left is 51.2 pounds squared.

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