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The question asks to identify the correct probability mass function (PMF) for the binomial distribution \( Bin(k | n, \theta) \).

The probability mass function (PMF) for a binomial distribution is given by the formula:

\[
P(k | n, \theta) = \binom{n}{k} \theta^k (1 - \theta)^{n - k}
\]

Where:
- \( n \) is the number of trials,
- \( k \) is the number of successes,
- \( \theta \) is the probability of success in a single trial,
- \( \binom{n}{k} \) is the binomial coefficient, which is \( \frac{n!}{k!(n - k)!} \).

Now, comparing this with the given options:
- (a) \( \binom{n}{k} \theta^k (1 - \theta)^{n - k} \) matches the correct PMF formula.

Thus, option **(a)** is the correct answer.

Would you like further details, or do you have any other questions?

Here are five related questions to deepen understanding:
1. What are the assumptions behind using a binomial distribution?
2. How do you compute the binomial coefficient \( \binom{n}{k} \)?
3. What are the mean and variance of a binomial distribution?
4. How would you compute the cumulative probability for a binomial distribution?
5. How does the binomial distribution relate to the Bernoulli distribution?

**Tip:** The binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success.

Which of the following represents the probability mass function (pmf) of binomial distribution Bin(k | n, θ)?

Explore how to identify the correct probability mass function (PMF) for the binomial distribution Bin(k | n, θ). The problem involves key concepts from probability and combinatorics, and uses the formula P(k | n, θ) = (n choose k) * θ^k * (1 - θ)^(n - k). Ideal for students at the undergraduate or advanced high school level.

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