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.