We are given that $X$ is a geometric random variable with the probability mass function (PMF):

$P(X = x) = (1 - p)p^x, \quad x = 0, 1, 2, \dots$ where $0 < p < 1$.

The goal is to find the moment generating function $G_X(t)$, which is defined as:

$G_X(t) = E(t^X) = \sum_{x=0}^{\infty} t^x P(X = x)$ Substitute the given PMF into this expression:

$G_X(t) = \sum_{x=0}^{\infty} t^x (1 - p) p^x$ Factor out $(1 - p)$ since it is constant:

$G_X(t) = (1 - p) \sum_{x=0}^{\infty} (tp)^x$ This is a geometric series with first term $1$ and ratio $tp$. The sum of an infinite geometric series $\sum_{x=0}^{\infty} r^x$ converges to $\frac{1}{1 - r}$ if $|r| < 1$.

Therefore, we require $|tp| < 1$ for the series to converge. This gives us the condition:

$|t| < \frac{1}{p}$

Now, applying the sum of the geometric series:

$G_X(t) = (1 - p) \frac{1}{1 - tp}$

Thus, the moment generating function is:

$G_X(t) = \frac{1 - p}{1 - tp}, \quad |t| < \frac{1}{p}$

Conclusion:

The moment generating function is $G_X(t) = \frac{1 - p}{1 - tp}$, and it is valid for $|t| < \frac{1}{p}$.

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Would you like further details on this solution or have any specific questions?

Here are 5 related questions to explore:

1. How do we compute moments from the moment generating function $G_X(t)$?
2. What is the expected value $E(X)$ for a geometric random variable?
3. Can you explain the significance of the condition $|t| < \frac{1}{p}$?
4. How would you compute the variance of $X$ using the moment generating function?
5. What are other applications of moment generating functions in probability theory?

Tip: The moment generating function can be used to calculate all moments (like mean and variance) by differentiating it with respect to $t$ and evaluating at $t = 0$.