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A probability mass function (PMF) is a fundamental concept when dealing with discrete random variables like our binomial variables here. Let's clarify what it means with binomial random variables. The PMF gives us the probability that a discrete random variable is exactly equal to some value.

For a binomial random variable, the PMF is determined by the number of trials (n) and the probability of success in each trial (p). It is given by:

• \( P(Y = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

This formula allows us to understand how likely it is for our binomial random variable to end with exactly \( k \) successes out of \( n \) trials.

Each term in the PMF formula has a special meaning:

• \( \binom{n}{k} \): The number of ways to choose \( k \) successes from \( n \) trials.
• \( p^k \): The probability of having \( k \) successes.
• \( (1-p)^{n-k} \): The probability of having \( n-k \) failures.

This approach allows us to precisely calculate the probabilities associated with different outcomes of a binomial trial, which can be quite useful in real-world applications.

Independent random variables are an important notion in probability theory. Two random variables are said to be independent if the occurrence of one event does not affect the probability of the other event occurring.

In our exercise, we have two such independent random variables, \( Y_1 \) and \( Y_2 \). Their independence allows specific simplifications when calculating complex scenarios like the sum \( Y_1 + n_2 - Y_2 \).

When we consider independent random variables, the joint probability of any two events occurring can be expressed as the product of their individual probabilities. This is a crucial property because it allows us to break down complex probabilities into more manageable pieces. For instance, for our variables:

• \( P(Y_1 = a, Y_2 = b) = P(Y_1 = a) \times P(Y_2 = b) \)

This property is leveraged when calculating the probability function for the composite random variable \( X = Y_1 + n_2 - Y_2 \), allowing us to separate and individually compute probabilities for \( Y_1 \) and \( Y_2 \) and then recombine them.

The law of total probability is a fundamental theorem that plays a vital role in solving probabilistic problems involving multiple scenarios or parts. In the case of our problem, this law helps us find the probability function of a variable formed by both \( Y_1 \) and \( Y_2 \).

This law allows us to express the probability of an event by considering all possible ways that the event can occur—breaking down complex, joint distributions into simpler calculations.

The sum formula used in our exercise stems from this law. It calculates the probability \( P(X = x) \) for the random variable \( X = Y_1 + n_2 - Y_2 \) by summing up probabilities over all possible values of one of the involved random variables:

• \( P(X = x) = \sum_{k_1=0}^{n_1} P(Y_1 = k_1) \cdot P(Y_2 = k_1 + n_2 - x) \)

This summation integrates probabilities of each possible outcome of \( Y_1 \), paired with the dependent outcome of \( Y_2 \), contributing to the final probability of \( X \). By using the law of total probability, we ensure that all potential combinations of \( Y_1 \) and \( Y_2 \) are considered, giving us a comprehensive picture of the probabilities at play.