Suppose X is a geometric random variable with X ∈ {0, 1, 2, . . .} and with
probability mass function
P(X = x) = (1 − p) px, x = 0, 1, 2, . . . ,
for some 0 < p < 1.
Find
GX (t) = E(tX )
and indicate for what values of t are appropriate.

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 \( t = 0 \).

Suppose X is a geometric random variable with X ∈ {0, 1, 2, . . .} and with probability mass function P(X = x) = (1 − p) px, x = 0, 1, 2, . . . , for some 0 < p < 1. Find GX(t) = E(t^X) and indicate for what values of t are appropriate.

Find the moment generating function GX(t) for a geometric random variable X with the given probability mass function P(X = x) = (1 − p) p^x. Understand the series convergence condition and how to apply the geometric series sum formula for |t| < 1/p. This solution is suitable for undergraduate probability courses.

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