Question 1177537
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!