### Probability and Expected Value of Storage Space for Upright FreezersAn appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let $X$ be the amount of storage space purchased by the next customer to buy a freezer. Suppose that $X$ has the probability mass function (pmf) given by the following table:| $ x $ | 13.5 | 15.9 | 19.1 ||-----------|------|------|------|| $ P(x) $| 0.2 | 0.4 | 0.4 |#### a. Compute $E(X)$The expected value $E(X)$ of a random variable $X$ with pmf $P(x)$ is calculated using the formula:\[ E(X) = \sum_{i} x_i P(x_i) \]Thus,\[ E(X) = (13.5 \times 0.2) + (15.9 \times 0.4) + (19.1 \times 0.4) \]#### b. Compute $E(X^2)$The expected value of $X^2$ is calculated using the formula:\[ E(X^2) = \sum_{i} x_i^2 P(x_i) \]Thus,\[ E(X^2) = (13.5^2 \times 0.2) + (15.9^2 \times 0.4) + (19.1^2 \times 0.4) \]#### c. Compute $V(X)$The variance $V(X)$ of a random variable $X$ is given by:\[ V(X) = E(X^2) - [E(X)]^2 \]

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An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has pmf 13.5 15.9 19.1 P(x) .2 .4 .4 a. Compute E(X) b. Compute E(X²) c. Compute V(X).

An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has pmf 13.5 15.9 19.1 P(x) .2 .4 .4 a. Compute E(X) b. Compute E(X²) c. Compute V(X).

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

### Probability and Expected Value of Storage Space for Upright Freezers

An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let $X$ be the amount of storage space purchased by the next customer to buy a freezer. Suppose that $X$ has the probability mass function (pmf) given by the following table:

| $ x $ | 13.5 | 15.9 | 19.1 |
|-----------|------|------|------|
| $ P(x) $| 0.2 | 0.4 | 0.4 |

#### a. Compute $E(X)$

The expected value $E(X)$ of a random variable $X$ with pmf $P(x)$ is calculated using the formula:

\[ E(X) = \sum_{i} x_i P(x_i) \]

Thus,

\[ E(X) = (13.5 \times 0.2) + (15.9 \times 0.4) + (19.1 \times 0.4) \]

#### b. Compute $E(X^2)$

The expected value of $X^2$ is calculated using the formula:

\[ E(X^2) = \sum_{i} x_i^2 P(x_i) \]

Thus,

\[ E(X^2) = (13.5^2 \times 0.2) + (15.9^2 \times 0.4) + (19.1^2 \times 0.4) \]

#### c. Compute $V(X)$

The variance $V(X)$ of a random variable $X$ is given by:

\[ V(X) = E(X^2) - [E(X)]^2 \]

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