SOLUTION: Find E(X) where X is the outcome when one rolls a six-sided balanced die. Find the mgf of X. Also, using the mgf of X, compute the variance of X Thank You

Question 1177292: Find E(X) where X is the outcome when one rolls a six-sided balanced die. Find the mgf of X. Also, using the mgf of X, compute the variance of X
Thank You
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**1. Find E(X)**
* X represents the outcome of rolling a six-sided balanced die.
* Possible values of X: 1, 2, 3, 4, 5, 6
* Probability of each outcome: 1/6
The expected value of X is:
* E(X) = Σ [x * P(X = x)]
* E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)
* E(X) = (1/6) * (1 + 2 + 3 + 4 + 5 + 6)
* E(X) = (1/6) * 21
* E(X) = 21/6 = 7/2 = 3.5
**2. Find the Moment Generating Function (MGF) of X**
The MGF of a discrete random variable X is:
* MX(t) = E(e^(tX)) = Σ [e^(tx) * P(X = x)]
In our case:
* MX(t) = (1/6) * (e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))
**3. Compute the Variance of X Using the MGF**
1. **Find MX'(t):**
* MX'(t) = (1/6) * (e^t + 2e^(2t) + 3e^(3t) + 4e^(4t) + 5e^(5t) + 6e^(6t))
2. **Find MX''(t):**
* MX''(t) = (1/6) * (e^t + 4e^(2t) + 9e^(3t) + 16e^(4t) + 25e^(5t) + 36e^(6t))
3. **Find E(X) and E(X²) using the MGF:**
* E(X) = MX'(0) = (1/6) * (1 + 2 + 3 + 4 + 5 + 6) = 21/6 = 7/2 = 3.5
* E(X²) = MX''(0) = (1/6) * (1 + 4 + 9 + 16 + 25 + 36) = 91/6
4. **Calculate Var(X):**
* Var(X) = E(X²) - [E(X)]²
* Var(X) = 91/6 - (7/2)²
* Var(X) = 91/6 - 49/4
* Var(X) = (182 - 147) / 12
* Var(X) = 35/12
**Answers**
* E(X) = 3.5
* MX(t) = (1/6) * (e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))
* Var(X) = 35/12

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