Problem 2 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 and 2, the outcomes and corresponding probability assignments for a discrete random variable \(X\) are listed. Draw the histogram for \(X\). Then find the expected value \(E(X)\), the variance \(\operatorname{Var}(X)\), and the standard deviation \(\sigma(X)\). $$ \begin{array}{l|c|c|c|c|c} \hline \text { Outcomes for } X & 0 & 2 & 4 & 6 & 8 \\ \hline \text { Probability } & \frac{1}{8} & \frac{3}{8} & \frac{1}{4} & \frac{1}{8} & \frac{1}{8} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Expected Value: 3, Variance: 6, Standard Deviation: \(\sqrt{6}\)

Step 1 - Construct the Histogram

First, draw the histogram for the discrete random variable (RV) \(X\) using the given outcomes and their corresponding probabilities. On the x-axis, label the outcomes: 0, 2, 4, 6, and 8. On the y-axis, mark the probabilities: \(\frac{1}{8}, \frac{3}{8}, \frac{1}{4}, \frac{1}{8}, \frac{1}{8}\). Create bars for each outcome with heights corresponding to their probabilities.

Step 2 - Calculate the Expected Value \(E(X)\)

The expected value \(E(X)\) can be found using the formula: \[ E(X) = \sum_{i} x_i P(X = x_i) \] Apply this to the provided data: \[ E(X) = 0 \cdot \frac{1}{8} + 2 \cdot \frac{3}{8} + 4 \cdot \frac{1}{4} + 6 \cdot \frac{1}{8} + 8 \cdot \frac{1}{8} \] Calculate each term and sum them.

Step 3 - Sum Up the Expected Value Calculation

Multiply each outcome by its respective probability and sum up the results: \[ E(X) = 0 \cdot \frac{1}{8} + 2 \cdot \frac{3}{8} + 4 \cdot \frac{1}{4} + 6 \cdot \frac{1}{8} + 8 \cdot \frac{1}{8} \] \[ = 0 + \frac{6}{8} + \frac{4}{4} + \frac{6}{8} + \frac{8}{8} \] \[ = 0 + \frac{6}{8} + 1 + \frac{6}{8} + 1 \] \[ = 3 \]

Step 4 - Calculate the Variance \(\operatorname{Var}(X)\)

The variance \(\operatorname{Var}(X)\) is calculated using the formula: \[ \operatorname{Var}(X) = E[(X - E(X))^2] \] This can be simplified as: \[ \operatorname{Var}(X) = \sum_{i} (x_i - E(X))^2 P(X = x_i) \] Apply \(E(X) = 3\) from the previous step and calculate the variance: \[ \operatorname{Var}(X) = \sum_{i} (x_i - 3)^2 P(X = x_i) \]

Step 5 - Sum Up the Variance Calculation

Calculate each term for the variance formula: \[ \operatorname{Var}(X) = (0 - 3)^2 \cdot \frac{1}{8} + (2 - 3)^2 \cdot \frac{3}{8} + (4 - 3)^2 \cdot \frac{1}{4} + (6 - 3)^2 \cdot \frac{1}{8} + (8 - 3)^2 \cdot \frac{1}{8} \] Simplify the expression: \[ = 9 \cdot \frac{1}{8} + 1 \cdot \frac{3}{8} + 1 \cdot \frac{1}{4} + 9 \cdot \frac{1}{8} + 25 \cdot \frac{1}{8} \] \[ = \frac{9}{8} + \frac{3}{8} + \frac{1}{4} + \frac{9}{8} + \frac{25}{8} \] \[ = \frac{9 + 3 + 2 + 9 + 25}{8} \] \[ = \frac{48}{8} \] \[ = 6 \]

Step 6 - Calculate the Standard Deviation \(\sigma(X)\)

The standard deviation \(\sigma(X)\) is the square root of the variance: \[ \sigma(X) = \sqrt{\operatorname{Var}(X)} = \sqrt{6} \]

Key Concepts

Discrete Random Variable

A discrete random variable (RV) is a type of random variable that can take on a finite or countable infinite set of values. These values are distinct and separate. For example, rolling a die has outcomes of 1, 2, 3, 4, 5, or 6. Similarly, in the given exercise, the RV \(X\) can take outcomes like 0, 2, 4, 6, and 8. Each outcome has a specific probability assigned to it.

Understanding discrete random variables helps in constructing probability distributions and performing further calculations such as the expected value (mean), variance, and standard deviation. Each outcome in a discrete random variable must have a probability between 0 and 1, and the sum of all probabilities must equal 1.

Expected Value

The expected value (\(E(X)\)) is a fundamental concept in probability, representing the average or mean of all possible outcomes of a discrete random variable, weighted by their probabilities. It's a measure of the central tendency. The formula to calculate the expected value is:

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

For the given exercise, we use the outcomes and their probabilities:

\[ E(X) = 0 \cdot \frac{1}{8} + 2 \cdot \frac{3}{8} + 4 \cdot \frac{1}{4} + 6 \cdot \frac{1}{8} + 8 \cdot \frac{1}{8} = 3 \]

This means that if we repeatedly chose numbers according to the probability distribution, we would expect the average to be 3 over time.

Histogram

A histogram is a graphical representation of the distribution of numerical data. It shows how often each different value in a set of data occurs, usually as vertical bars. In the context of discrete random variables, a histogram helps visualize the probability distribution.

To draw the histogram for the discrete random variable \(X\), follow these steps:

• On the x-axis, label the outcomes (0, 2, 4, 6, 8).
• On the y-axis, label the probabilities (\(\frac{1}{8}\), \(\frac{3}{8}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{8}\)).
• For each outcome, draw a vertical bar with a height corresponding to its probability.

This visual representation makes it easier to see the distribution and compare the likelihood of different outcomes.

Probability Distribution

A probability distribution describes how the values of a random variable are distributed. For a discrete random variable, it provides the probabilities of the possible outcomes. These probabilities must add up to 1.

In the given exercise, the probability distribution for \(X\) is:

• \(P(X = 0) = \frac{1}{8} = 0.125\)
• \(P(X = 2) = \frac{3}{8} = 0.375\)
• \(P(X = 4) = \frac{1}{4} = 0.25\)
• \(P(X = 6) = \frac{1}{8} = 0.125\)
• \(P(X = 8) = \frac{1}{8} = 0.125\)

This distribution tells us how likely each outcome is when considering the random variable \(X\). Knowing the probability distribution is essential for calculating other statistical measures like the expected value, variance, and standard deviation.