This image provides the definition of a generating function for a discrete random variable $X$ with a given probability distribution. Let's break it down and explain step by step:

Definition Recap:

1. Random Variable $X$: $X$ is a discrete random variable taking integer values.

2. Probability Distribution $p_k$: $p_k = P(X = k)$ is the probability that $X$ takes the value $k$.

3. Generating Function $G_X(t)$: The generating function $G_X(t)$ is defined as: [ G_X(t) = \mathbb{E}[t^X] = \sum_{k=0}^\infty t^k p_k, \quad \forall t \in [-1, 1]. ] Here:

   • $\mathbb{E}[t^X]$ is the expected value of $t^X$.
   • $t^k p_k$ represents the contribution of each probability $p_k$ scaled by $t^k$.

Explanation of Key Points:

1. Purpose: The generating function compactly encodes the probability distribution $p_k$. Each coefficient in the power series of $G_X(t)$ corresponds to a probability.

2. Range of $t$: The function $G_X(t)$ is valid for $t \in [-1, 1]$ because the series converges in this range.

3. Properties:

   • The value of $G_X(1)$ gives the sum of all probabilities, which is $1$ (since it's a valid probability distribution).
   • Derivatives of $G_X(t)$ can be used to compute moments of the distribution.

Example:

Let $X$ be a discrete random variable that represents the result of a fair six-sided die roll. The probabilities are: