Let's solve the questions step by step.

Question 2

We are tasked with expressing $\mathbb{E}[f(T)]$ as an integral over $\Omega$. Here is the step-by-step approach:

1. Understanding the setup:
   $T: (E, \mathcal{E}) \to (F, \mathcal{F})$ is a measurable mapping.
   $f: (F, \mathcal{F}) \to ([0, \infty], \mathcal{B}(\mathbb{R}))$ is a measurable function.

   The composition $f(T): \Omega \to \mathbb{R}_+$ is measurable, allowing us to define expectations.

2. Expected value over $\Omega$:
   By definition, the expected value of $f(T)$ is given by: $\mathbb{E}[f(T)] = \int_\Omega f(T(\omega)) \, d\mathbb{P}(\omega),$ where $\mathbb{P}$ is the probability measure on $\Omega$.

3. Expressing using change of measure:
   $T$ induces a pushforward measure $\mathbb{P}_Y = \mathbb{P} \circ Y^{-1}$ on $(E, \mathcal{E})$, which is the distribution of $T$. Using the change of variables formula: $\mathbb{E}[f(T)] = \int_E f(y) \, d\mathbb{P}_Y(y),$ where $y = T(\omega)$ under the mapping.

   This equation expresses $\mathbb{E}[f(T)]$ as an integral over $E$.

Question 3

We are tasked with proving that: $\sum_{i=1}^n X_i \sim \text{Bin}(nk, p),$ where $X_1, \dots, X_n$ are i.i.d. random variables, each $X_i \sim \text{Bin}(k, p)$.

1. Definition of $X_i$:
   Each $X_i \sim \text{Bin}(k, p)$ means that $X_i$ counts the number of successes in $k$ independent trials, each with success probability $p$. The probability mass function is: $\mathbb{P}(X_i = m) = \binom{k}{m} p^m (1-p)^{k-m}, \quad m \in \{0, 1, \dots, k\}.$

2. Sum of independent binomial variables:
   The sum $S = \sum_{i=1}^n X_i$ represents the total number of successes in $nk$ trials. Since the trials across all $X_i$ are independent, $S$ follows a Binomial distribution: $S \sim \text{Bin}(nk, p).$

3. Justification using properties of Binomial distributions:

   • The total number of trials is $n \times k$, because each $X_i$ contributes $k$ trials.
   • The probability of success $p$ remains unchanged since each trial is independent.
   • Therefore, the sum of $n$ independent $\text{Bin}(k, p)$ random variables is $\text{Bin}(nk, p)$.

4. Hint application ($\Omega = \{0, 1\}^{nk}$):
   Using $\Omega = \{0, 1\}^{nk}$, we can interpret each trial as an indicator variable over $\Omega$. The sum $\sum_{i=1}^n X_i$ aggregates the results over all $nk$ trials, confirming that $S \sim \text{Bin}(nk, p)$.

Do you need further clarifications or more detailed steps? Here are 5 related questions to explore:

1. How does the pushforward measure $\mathbb{P}_Y$ simplify calculations in measure theory?
2. Can you provide another example of changing variables in integrals with a stochastic mapping?
3. What are the key properties of a Binomial distribution?
4. How does the Central Limit Theorem relate to summing independent Binomial variables?
5. What happens if the success probability $p$ is different for each random variable $X_i$?

Tip: When dealing with sums of independent random variables, always check if they share the same distribution type to simplify computations.