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### Problem Outline:
Given an experiment with probability \( p \) of success and \( 1 - p \) of failure, repeated \( n \) times with independent trials, we are dealing with a **binomial distribution**. Let \( X \) represent the number of successes among the \( n \) trials.

The probability mass function (PMF) for \( X \) is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}, \quad k = 0, 1, 2, \dots, n.
\]

Random variables with this PMF are called **binomial random variables** with parameters \( n \) and \( p \).

#### Part (a): Verify the following identity:
\[
\sum_{k=0}^{n} P(X = k) = 1
\]
This shows that the total probability across all possible outcomes \( k \) must sum to 1. The hint suggests using the **Binomial Theorem** for verification.

#### Part (b): Compute the expected value \( E[X] \).

We are given a hint that:
\[
k \binom{n}{k} = n \binom{n-1}{k-1}.
\]
This suggests we use the properties of binomial coefficients to simplify the computation of the expected value.

#### Part (c): Compute the variance \( \text{Var}[X] \).

### Step-by-step solutions:

#### (a) Verifying \( \sum_{k=0}^{n} P(X = k) = 1 \)
We need to prove that the sum of the binomial probabilities across all \( k \) from 0 to \( n \) equals 1. Using the PMF of the binomial distribution:

\[
\sum_{k=0}^{n} \binom{n}{k} p^k (1 - p)^{n-k}.
\]

This is exactly the **Binomial Theorem**, which states that:

\[
(p + (1 - p))^n = 1^n = 1.
\]

Thus, the sum of all binomial probabilities equals 1, as required.

#### (b) Compute \( E[X] \)
The expected value of a binomial random variable \( X \) with parameters \( n \) and \( p \) is known to be:

\[
E[X] = n p.
\]

To derive this, we use the property of linearity of expectation. Specifically, the binomial random variable can be seen as the sum of \( n \) independent Bernoulli trials, where each trial has an expected value \( p \). Therefore:

\[
E[X] = \sum_{k=0}^{n} k \cdot P(X = k).
\]

Using the given hint that:

\[
k \binom{n}{k} = n \binom{n-1}{k-1},
\]

we can simplify the sum, eventually arriving at:

\[
E[X] = n p.
\]

#### (c) Compute \( \text{Var}[X] \)
The variance of a binomial random variable \( X \) is:

\[
\text{Var}(X) = n p (1 - p).
\]

This result can be derived using the fact that \( \text{Var}(X) = E[X^2] - (E[X])^2 \), and utilizing the properties of binomial random variables.

### Summary:
- **(a)**: The sum of binomial probabilities equals 1, by the Binomial Theorem.
- **(b)**: The expected value \( E[X] = n p \).
- **(c)**: The variance \( \text{Var}[X] = n p (1 - p) \).

Would you like a more detailed explanation for any of these parts?

Here are 5 follow-up questions related to this problem:
1. How does the binomial theorem help simplify the sum in part (a)?
2. What is the significance of the expected value in the context of binomial random variables?
3. Can you derive the formula for variance using moment generating functions?
4. How would the expected value and variance change if \( p = \frac{1}{2} \)?
5. How would you compute higher moments (like skewness or kurtosis) for the binomial distribution?

**Tip**: Remember that for binomial distributions, you can always interpret the expected value \( E[X] \) as the product of the number of trials and the probability of success per trial.

Let 0 < p < 1 and n = 1, 2, 3, ... A certain experiment has a probability p of success and a probability 1 - p of failure. The experiment is repeated n times, with all experiments independent. Let X be the number of successes among the n trials. Verify that the sum of probabilities of the binomial distribution equals 1, compute the expected value E[X], and compute the variance Var[X].

Explore the properties of the binomial distribution, including verification of probabilities, computation of the expected value, and variance. Learn how to apply the Binomial Theorem in the context of probability. Suitable for college-level or advanced high school students.

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