A discrete random variable X has the following probability mass function (PMF): x 1 2 3 4 Pr(x) 0.1 0.3 0.4 0.2.

For the discrete random variable \( X \) with the given probability mass function (PMF), let's organize the data first:

| \( x \)    | 1   | 2   | 3   | 4   |
|------------|-----|-----|-----|-----|
| \( P(X=x) \) | 0.1 | 0.3 | 0.4 | 0.2 |

### Key concepts related to a discrete random variable like this include:
1. **Expected value (mean)**:  
   The expected value \( E(X) \) is calculated as:
   \[
   E(X) = \sum_{x} x \cdot P(X = x)
   \]
   Let's compute this.

2. **Variance**:  
   The variance \( \text{Var}(X) \) measures the spread of the values around the mean and is calculated as:
   \[
   \text{Var}(X) = E(X^2) - (E(X))^2
   \]
   where \( E(X^2) \) is the expected value of \( X^2 \), given by:
   \[
   E(X^2) = \sum_{x} x^2 \cdot P(X = x)
   \]

3. **Standard deviation**:  
   This is the square root of the variance:
   \[
   \sigma_X = \sqrt{\text{Var}(X)}
   \]

I'll now compute the **expected value**, **variance**, and **standard deviation** for \( X \).Here are the computed values for the discrete random variable \( X \):

1. **Expected value \( E(X) \)**: 2.7
2. **Variance \( \text{Var}(X) \)**: 0.81
3. **Standard deviation \( \sigma_X \)**: 0.9

Would you like further details on these calculations or have any questions?

### Related questions:
1. How is the expected value useful in understanding random variables?
2. What does the variance tell us about the distribution of \( X \)?
3. How does standard deviation relate to variance in discrete distributions?
4. How can we calculate the skewness of a discrete random variable?
5. What is the interpretation of the PMF graph for a discrete random variable?

### Tip:
The expected value provides the long-term average of a random variable if the experiment is repeated many times.

This page provides a detailed solution for a discrete random variable X with a given probability mass function (PMF). Learn how to compute the expected value, variance, and standard deviation for X using key probability formulas. Suitable for Grades 10-12 students.

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