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The problem asks for the **Moment Generating Function (MGF)** of the discrete random variable \( X \) that takes values \( 1 \) and \( -1 \) with probabilities \( \frac{1}{2} \) each.

The MGF of a discrete random variable \( X \) is given by:

\[
M_X(t) = \mathbb{E}[e^{tX}] = \sum_{x} e^{tx} P(X = x)
\]

For this particular random variable:

- \( X = 1 \) with probability \( \frac{1}{2} \),
- \( X = -1 \) with probability \( \frac{1}{2} \).

Thus, the MGF \( M_X(t) \) becomes:

\[
M_X(t) = e^{t \cdot 1} \cdot \frac{1}{2} + e^{t \cdot (-1)} \cdot \frac{1}{2}
\]

Simplifying:

\[
M_X(t) = \frac{1}{2} e^t + \frac{1}{2} e^{-t}
\]

This matches option **(C)**.

### Conclusion:
The correct answer is **(C) \( \frac{e^t + e^{-t}}{2} \)**.

Would you like more details or have any further questions?

### Related Questions:
1. What is the MGF formula for a continuous random variable?
2. How do you derive the mean and variance from the MGF?
3. What are the properties of moment generating functions?
4. How does the MGF relate to the characteristic function of a random variable?
5. How can the MGF be used to determine moments (e.g., mean, variance) of a distribution?

### Tip:
To find moments like the mean and variance from an MGF, differentiate it with respect to \( t \) and evaluate at \( t = 0 \).

What is the Moment generating function for the discrete random variable X, where X = 1 with probability 1/2 and X = -1 with probability 1/2?

Learn how to find the moment generating function (MGF) for the discrete random variable X, where X takes the values 1 and -1 with equal probabilities. This problem involves understanding the MGF formula and applying it to a simple discrete distribution.

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