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The expected value, often denoted as \(E(X)\), is a fundamental concept in probability theory. It represents the average outcome if an experiment is repeated a large number of times. In the context of a negative binomial distribution, the expected value gives the average number of failures before a specified number of successes occurs.

For a negative binomial distribution, the expected value is calculated using the formula:

• \(E(X) = \frac{r(1-p)}{p}\)

Where \(r\) is the number of successes, and \(p\) is the probability of success on each trial. In this exercise, \(p=0.2\) and \(r=4\). Plugging these values into the formula, we achieve:

• \(E(X) = \frac{4(1-0.2)}{0.2} = 16\)

This means, on average, 16 failures occur before 4 successes are reached in this scenario. Understanding the expected value assists in predicting outcomes and planning around uncertainty.

The probability mass function (PMF) is crucial for discrete random variables like those in the negative binomial distribution. It tells us the probability that a random variable will take on a specific value. The PMF for a negative binomial distribution is:

• \(P(X=k) = \binom{k+r-1}{r-1} p^r (1-p)^{k}\)

Here, \(X\) is the number of failures before the \(r\)-th success, \(p\) is the probability of success, and \(\binom{k+r-1}{r-1}\) is a binomial coefficient. This expression gives us the probability of encountering exactly \(k\) failures.

For instance, to find \(P(X=20)\) with \(p=0.2\) and \(r=4\), we compute:

• \(P(X=20) = \binom{23}{3} (0.2)^4 (0.8)^{20} \approx 0.03208\)

This probability detail is essential for understanding the likelihood of different outcomes in a sequence of trials.

The mode of a distribution is the value that appears most frequently. For the negative binomial distribution, identifying the mode is crucial since it represents the most likely number of failures before the specified number of successes is achieved.

To determine the mode, one often looks at the probabilities around the expected value calculated using the PMF. In this exercise, we compare \(P(X=19)\), \(P(X=20)\), and \(P(X=21)\). Calculations show:

• \(P(X=19) \approx 0.035395\)
• \(P(X=20) \approx 0.03208\)
• \(P(X=21) \approx 0.029825\)

Here, \(X=19\) exhibits the highest probability. Therefore, it is the mode of this negative binomial distribution. Recognizing the mode helps in making predictive statements about the most probable outcomes.

The binomial coefficient, represented as \(\binom{n}{k}\), is a mathematical concept used in combinatorics to count the number of ways to choose \(k\) successes from \(n\) trials without regard to the order of selection. It appears frequently in probability calculations, especially in distributions like the binomial and negative binomial.

This coefficient is calculated as:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

In the context of the negative binomial distribution, it is used to determine probabilities such as \(P(X=k) = \binom{k+r-1}{r-1} p^r (1-p)^{k}\). For example:

• For \(P(X=20)\), we calculate \(\binom{23}{3} = 1771\), which is a crucial part of determining the overall probability.

Understanding binomial coefficients provides a foundation for working with various probability distributions and solving problems involving selection and arrangement.