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 so much

Question 1177537: 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 so much
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
Absolutely! Let's break down the calculations step by step.
1. **Expected Value (E(X))**
The expected value of a discrete random variable is the sum of the products of each possible value and its probability. For a fair six-sided die, each outcome (1 to 6) has a probability of 1/6. Therefore:
```
E(X) = (1)(1/6) + (2)(1/6) + (3)(1/6) + (4)(1/6) + (5)(1/6) + (6)(1/6)
= 21/6 = 3.5
```
So, the expected value of a single roll is 3.5.
2. **Moment Generating Function (MGF)**
The moment generating function of a random variable X is defined as:
```
M_X(t) = E(e^(tX))
```
For our die roll:
```
M_X(t) = E(e^(tX)) = (1/6)(e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))
```
3. **Variance using the MGF**
The variance can be computed using the MGF as follows:
```
Var(X) = E(X^2) - (E(X))^2
```
We can find E(X^2) using the second derivative of the MGF:
```
E(X^2) = M_X''(0)
```
Taking the second derivative of M_X(t) and evaluating it at t=0 is a bit tedious, but doing the calculation gives us:
```
E(X^2) = 91/6
```
Now we can compute the variance:
```
Var(X) = E(X^2) - (E(X))^2 = (91/6) - (3.5)^2 = 35/12
```
Feel free to ask if you have any further questions or would like to explore other properties of the die roll distribution!
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