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

Question

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

UpStudy AI · Accepted Answer

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!
The expected value of $$ X $$ is \$10.93, with a variance of 210.73 and a standard deviation of approximately \$14.52.

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