Suppose X is a Poisson random variable; so it is discrete and takes values on
{0, 1, 2, . . .} and the probability mass function is
P (X = x) = λx
x! e−λ
for some λ > 0.
Find the generating function for X given by
GX (t) = E(tX ).

Given that \( X \) is a Poisson random variable with parameter \( \lambda \), its probability mass function (PMF) is:

\[
P(X = x) = \frac{\lambda^x}{x!} e^{-\lambda}, \quad x = 0, 1, 2, \dots
\]

### Step 1: Definition of the Generating Function

The generating function \( G_X(t) \) is defined as the expected value of \( t^X \), i.e.,

\[
G_X(t) = E(t^X) = \sum_{x=0}^{\infty} t^x P(X = x)
\]

Substituting the PMF of \( X \) into this expression:

\[
G_X(t) = \sum_{x=0}^{\infty} t^x \frac{\lambda^x}{x!} e^{-\lambda}
\]

### Step 2: Factor out constants

We can factor out \( e^{-\lambda} \), which does not depend on \( x \):

\[
G_X(t) = e^{-\lambda} \sum_{x=0}^{\infty} \frac{(\lambda t)^x}{x!}
\]

### Step 3: Recognize the power series

The remaining sum is a well-known power series expansion of the exponential function:

\[
\sum_{x=0}^{\infty} \frac{(\lambda t)^x}{x!} = e^{\lambda t}
\]

### Step 4: Combine terms

Thus, the generating function becomes:

\[
G_X(t) = e^{-\lambda} \cdot e^{\lambda t} = e^{\lambda (t - 1)}
\]

### Final Answer:

The generating function for the Poisson random variable \( X \) is:

\[
G_X(t) = e^{\lambda (t - 1)}
\]

Would you like further clarification on any step?

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Here are five related questions to consider:
1. How does the generating function for the Poisson distribution compare to other discrete distributions?
2. What is the purpose of a generating function in probability theory?
3. Can you compute the mean and variance of \( X \) using the generating function?
4. How would the generating function change if the Poisson parameter \( \lambda \) varies?
5. How can generating functions help in solving recurrence relations?

**Tip:** The generating function is particularly useful in finding moments of a random variable, as derivatives of the generating function at \( t = 1 \) provide these moments.

Suppose X is a Poisson random variable; so it is discrete and takes values on {0, 1, 2, . . .} and the probability mass function is P(X = x) = λ^x / x! * e^(-λ) for some λ > 0. Find the generating function for X given by GX(t) = E(t^X).

Explore how to find the generating function for a Poisson random variable. This problem covers key probability theory concepts, including the Poisson distribution and its generating function, using exponential series expansion. Suitable for undergraduate-level students.

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