## Discrete Random Variable Probability Distribution### Problem StatementThe discrete random variable $X$ has a probability distribution given by:\[ P(X = x) = \frac{4x + 7}{68} \quad \text{for} \; x = 1, 2, 3, 4. \]### Questions1. **Find:** 1. **(a)** $ E(X) $ 2. **(b)** $ E(X^2) $ 3. **(c)** $ E(X^2 + 5X - 2) $2. **Verify** that:\[ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2. \]### Explanation1. **Expected Value ($ E(X) $):**The expected value, $ E(X) $, of a discrete random variable is the sum of the product of each possible value of the variable and its corresponding probability.2. **Expected Value of $ X^2 $ ( $ E(X^2) $):**The expected value of $ X^2 $ is calculated by summing the products of each possible value squared, and its corresponding probability.3. **Expected Value of the Expression ( $ E(X^2 + 5X - 2) $):**For a linear combination of expected values, the expected value of the combined expression can be separated into the sum of the expected values of each term, each appropriately weighted.### Verifying the Formula:To verify $ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2 $, we can use the properties of expected values, which states that:\[ E(aX + bY + c) = aE(X) + bE(Y) + c \]where $a$, $b$, and $c$ are constants and $X$ and $Y$ are random variables. This property can be used to simplify and verify the given expression.

Answered: Image /qna-images/answer/3c93a51b-9fa4-4339-931a-00e729b9a4d7.jpg

7 The discrete random variable X has probability distribution given by (4x +7) P(X= x) = for x= 1, 2, 3, 4. 68 (i) Find (a) E(X) (b) E(X²) (c) E(X²+5X– 2). (ii) Verify that E(X² + 5X– 2) = E(X²) + 5E(X) – 2.

7 The discrete random variable X has probability distribution given by (4x +7) P(X= x) = for x= 1, 2, 3, 4. 68 (i) Find (a) E(X) (b) E(X²) (c) E(X²+5X– 2). (ii) Verify that E(X² + 5X– 2) = E(X²) + 5E(X) – 2.

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## Discrete Random Variable Probability Distribution

### Problem Statement
The discrete random variable $X$ has a probability distribution given by:

\[ P(X = x) = \frac{4x + 7}{68} \quad \text{for} \; x = 1, 2, 3, 4. \]

### Questions
1. **Find:**
1. **(a)** $ E(X) $
2. **(b)** $ E(X^2) $
3. **(c)** $ E(X^2 + 5X - 2) $

2. **Verify** that:

\[ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2. \]

### Explanation

1. **Expected Value ($ E(X) $):**
The expected value, $ E(X) $, of a discrete random variable is the sum of the product of each possible value of the variable and its corresponding probability.

2. **Expected Value of $ X^2 $ ( $ E(X^2) $):**
The expected value of $ X^2 $ is calculated by summing the products of each possible value squared, and its corresponding probability.

3. **Expected Value of the Expression ( $ E(X^2 + 5X - 2) $):**
For a linear combination of expected values, the expected value of the combined expression can be separated into the sum of the expected values of each term, each appropriately weighted.

### Verifying the Formula:
To verify $ E(X^2 + 5X - 2) = E(X^2) + 5E(X) - 2 $, we can use the properties of expected values, which states that:

\[ E(aX + bY + c) = aE(X) + bE(Y) + c \]

where $a$, $b$, and $c$ are constants and $X$ and $Y$ are random variables. This property can be used to simplify and verify the given expression.

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