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The problem involves calculating the mean, variance, and standard deviation of a discrete random variable \( X \) from a given probability distribution.

The table provided shows the values of \( X \) and their corresponding probabilities \( P(X) \):

\[
\begin{array}{c|c|c|c|c|c}
X & 2 & 5 & 6 & 13 & 18 \\
P(X) & 0.11 & 0.09 & 0.11 & 0.14 & 0.55 \\
\end{array}
\]

### Step 1: Calculate the mean (\(\mu\)) or expected value \(E(X)\)
The mean is the expected value, calculated as:

\[
\mu = E(X) = \sum X_i \cdot P(X_i)
\]

### Step 2: Calculate the variance (\(\sigma^2\))
Variance is calculated as:

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

where

\[
E(X^2) = \sum X_i^2 \cdot P(X_i)
\]

### Step 3: Calculate the standard deviation (\(\sigma\))
The standard deviation is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

I'll perform the calculations for you.Here are the calculated values based on the given dataHere are the calculated values based on the given 

- The mean (expected value) \( \mu \) is approximately \( 13.05 \).
- The variance \( \sigma^2 \) is approximately \( 38.2075 \).
- The standard deviation \( \sigma \) is approximately \( 6.1812 \).

Would you like more details or have any questions?

Here are some related questions to further expand on the topic:
1. How is the expected value of a discrete random variable different from a continuous one?
2. Can variance ever be negative? Why or why not?
3. What is the significance of standard deviation in a probability distribution?
4. How does changing the probability distribution affect the variance?
5. How would you calculate the mean and variance if the values of \( X \) were weighted differently?

**Tip:** Remember, the expected value is a measure of central tendency, while variance and standard deviation give insights into how spread out the values are from the mean.

Consider the discrete random variable X given in the table. Calculate the mean, variance, and standard deviation of X. Round to 4 decimals.

Learn how to calculate the mean, variance, and standard deviation of a discrete random variable X from a probability distribution. This problem provides a step-by-step guide, making it ideal for students in Grades 10-12.

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