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.