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The Probability Mass Function (PMF) is a key concept when dealing with discrete random variables like those in a binomial distribution. It describes the probability that a discrete random variable is exactly equal to some value. In our example, the random variable is the number of credit card holders that will default. Each outcome, like exactly 3 defaults, can be computed using the PMF.

In the context of a binomial distribution, the PMF is given by the formula: \[ P(X = k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k} \]Here,

• \( P(X = k) \) is the probability of exactly \( k \) successes (e.g., defaults),
• \( C(n, k) \) is the number of combinations of \( n \) items taken \( k \) at a time,
• \( p^k \) is the probability of \( k \) successes,
• \( (1-p)^{n-k} \) is the probability of all other \( n-k \) outcomes being failures.

This function helps determine the likelihood of various outcomes in the exercise.

The Combination Function, often represented as \( C(n, k) \), is crucial in calculating probabilities in binomial distributions. It determines the number of ways to choose \( k \) successes (defaults, in our case) out of \( n \) trials (the credit cards issued).

Mathematically, it is represented as:\[C(n, k) = \frac{n!}{k! (n-k)!}\]where \(!\) denotes a factorial, a product of all positive integers up to that number.

For our example:

• To find \( C(12, 3) \), we calculate \( \frac{12!}{3! \cdot 9!} \), determining the ways to choose 3 defaults out of 12 issued cards.
• Similarly, \( C(12, 1) \) is calculated as \( \frac{12!}{1! \cdot 11!} \) for exactly 1 default.
• \( C(12, 0) \) is \( \frac{12!}{0! \cdot 12!} \), representing the scenario where none default.

This function is a building block in evaluating probabilities for each scenario outlined in probability mass function calculations.

A Random Variable is a fundamental concept in probability and statistics. It is a variable whose possible values are outcomes of a random phenomenon. In this specific exercise, the random variable \( X \) represents the number of customers who default on their credit card debt.

Two main characteristics of a random variable:

• Discrete: Values are countable as they come from a finite set. For example, the count of defaulting customers can be 0, 1, 2, up to 12 if all defaulted.
• Continuous: Cannot be applied here but relevant when dealing with ranges and non-countable outcomes.

Binomial random variables like \( X \) have two possible outcomes for each trial – success (default) or failure (no default). Thus, the binomial distribution aptly models the situation, providing probabilities for each possible value of \( X \). This helps evaluate real-world scenarios, just as with the credit card defaults.

The Probability of Default is a specific probability that informs us about the chances of a credit card holder failing to fulfill their debt obligations. In the exercise, this probability is given as \( 0.08 \) or 8%.

This particular probability is used as a 'success' probability in binomial distribution because we are counting how many defaults (successes) occur among the issued credit cards.

• The probability mass function uses this as part of its calculation for each of the cases – 3 defaults, 1 default, and no defaults.
• This probability must be constant for the binomial distribution to be applicable.

To calculate, this potential outcome probability is raised to the power corresponding to the number of defaults and then multiplied by the inverse probability for non-default cases. Understanding this concept allows us to anticipate the financial impacts related to customer defaults effectively.