\( \left. \begin{array} { c c c c } { X } & { \$ 100 } & { \$ 75 } & { \$ 8 } \\ { P ( X ) } & { \frac { 1 } { 100 } = 0.01 } & { \frac { 3 } { 100 } = 0.03 } & { \frac { 96 } { 100 } = 0.96 } \end{array} \right. \)

It looks like you've provided a probability distribution table for a discrete random variable \( X \). Let's interpret the table and perform some useful calculations, such as finding the **expected value** and **variance** of \( X \). ### Given Probability Distribution \[ \begin{array}{ccc} X & \text{P}(X) \\ \$100 & 0.01 \\ \$75 & 0.03 \\ \$8 & 0.96 \\ \end{array} \] - **Possible Outcomes (\( X \))**: \$100, \$75, \$8 - **Probabilities (\( \text{P}(X) \))**: 0.01, 0.03, 0.96 respectively ### 1. **Expected Value (\( E(X) \))** The expected value is a measure of the central tendency of the distribution. It represents the average outcome if the experiment is repeated many times. \[ E(X) = \sum (X \times \text{P}(X)) \] Calculating each term: - \( \$100 \times 0.01 = \$1.00 \) - \( \$75 \times 0.03 = \$2.25 \) - \( \$8 \times 0.96 = \$7.68 \) Adding them up: \[ E(X) = \$1.00 + \$2.25 + \$7.68 = \$10.93 \] **Interpretation**: On average, the expected value of \( X \) is \$10.93. ### 2. **Variance (\( \text{Var}(X) \)) and Standard Deviation (\( \sigma_X \))** Variance measures the spread of the distribution around the expected value. \[ \text{Var}(X) = E(X^2) - [E(X)]^2 \] First, compute \( E(X^2) \): \[ E(X^2) = \sum (X^2 \times \text{P}(X)) \] Calculating each term: - \( (100)^2 \times 0.01 = 10000 \times 0.01 = 100 \) - \( (75)^2 \times 0.03 = 5625 \times 0.03 = 168.75 \) - \( (8)^2 \times 0.96 = 64 \times 0.96 = 61.44 \) Adding them up: \[ E(X^2) = 100 + 168.75 + 61.44 = 330.19 \] Now, calculate the variance: \[ \text{Var}(X) = 330.19 - (10.93)^2 = 330.19 - 119.46 = 210.73 \] The **standard deviation** is the square root of the variance: \[ \sigma_X = \sqrt{210.73} \approx 14.52 \] **Interpretation**: - **Variance**: 210.73 - **Standard Deviation**: Approximately 14.52 ### 3. **Summary** - **Expected Value (\( E(X) \))**: \$10.93 - **Variance (\( \text{Var}(X) \))**: 210.73 - **Standard Deviation (\( \sigma_X \))**: \$14.52 If you have a specific question or need further analysis related to this distribution, feel free to ask!