# Expectation and Variance of a Random Variable## RANDOM VARIABLES AND DISTRIBUTIONS### Expectation and variance of a random variableLet $ X $ be a random variable with the following probability distribution.| Value $ x $ of $ X $ | $ P(X = x) $ ||--------------------------|----------------|| -1 | 0.35 || 0 | 0.05 || 1 | 0.45 || 2 | 0.15 |Complete the following. (If necessary, consult a [list of formulas](#)).**(a) Find the expectation $ E(X) $ of $ X $.**\[ E(X) = 1.1 \]**(b) Find the variance $ Var(X) $ of $ X $.**\[ Var(X) = \quad \]*Buttons for Explanation and Check are provided.*---### Explanation of Terms and Concepts:**Expectation (Mean) $ E(X) $:**The expectation $ E(X) $ of a random variable $ X $ is a measure of the central tendency. It is computed as:\[ E(X) = \sum_{i} x_i \cdot P(X = x_i) \]**Variance $ Var(X) $:**The variance $ Var(X) $ measures the spread of the random variable values around the mean. It is computed as:\[ Var(X) = \sum_{i} (x_i - E(X))^2 \cdot P(X = x_i) \]### Steps to Compute:**Expectation:**1. Multiply each value $ x_i $ by its probability $ P(X = x_i) $.2. Sum all these products to get $ E(X) $.**Variance:**1. Compute the mean $ E(X) $.2. Subtract the mean from each $ x_i $ to get the difference.3. Square each difference.4. Multiply each squared difference by the corresponding probability $ P(X = x_i) $.5. Sum these products to get $ Var(X) $. These calculations help in understanding the distribution and spread of the random variable $ X $, which is crucial in statistical analysis and prediction models.

Given data, a)Expectation E(X)=? b)Variance V(X)=?

Answered: Let X be a random variable with the…

Let X be a random variable with the following probability distribution. Value x of X P(X=x) -1 0.35 0.05 1. 0.45 2. 0.15

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# Expectation and Variance of a Random Variable

## RANDOM VARIABLES AND DISTRIBUTIONS
### Expectation and variance of a random variable

Let $ X $ be a random variable with the following probability distribution.

| Value $ x $ of $ X $ | $ P(X = x) $ |
|--------------------------|----------------|
| -1 | 0.35 |
| 0 | 0.05 |
| 1 | 0.45 |
| 2 | 0.15 |

Complete the following. (If necessary, consult a [list of formulas](#)).

**(a) Find the expectation $ E(X) $ of $ X $.**

\[ E(X) = 1.1 \]

**(b) Find the variance $ Var(X) $ of $ X $.**

\[ Var(X) = \quad \]

*Buttons for Explanation and Check are provided.*

---

### Explanation of Terms and Concepts:

**Expectation (Mean) $ E(X) $:**
The expectation $ E(X) $ of a random variable $ X $ is a measure of the central tendency. It is computed as:
\[ E(X) = \sum_{i} x_i \cdot P(X = x_i) \]

**Variance $ Var(X) $:**
The variance $ Var(X) $ measures the spread of the random variable values around the mean. It is computed as:
\[ Var(X) = \sum_{i} (x_i - E(X))^2 \cdot P(X = x_i) \]

### Steps to Compute:

**Expectation:**
1. Multiply each value $ x_i $ by its probability $ P(X = x_i) $.
2. Sum all these products to get $ E(X) $.

**Variance:**
1. Compute the mean $ E(X) $.
2. Subtract the mean from each $ x_i $ to get the difference.
3. Square each difference.
4. Multiply each squared difference by the corresponding probability $ P(X = x_i) $.
5. Sum these products to get $ Var(X) $.

These calculations help in understanding the distribution and spread of the random variable $ X $, which is crucial in statistical analysis and prediction models.

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