What is the probability density function for the negative binomial distribution?

What is the probability density function for the negative binomial distribution?

• A. f(x) = (r + x - 1)!/x!(r-1)! * p^r * q^x
• B. f(x) = p^x * q
• C. f(x) = (1 - p)^x * p
• D. f(x) = e^(-λ) * λ^x / x!

Answer: (A)

The probability density function for the negative binomial distribution describes the probability of obtaining a specified number of successes in a series of independent Bernoulli trials, where the trials are continued until a predefined number of successes occurs. In the formula provided, \( f(x) = \frac{(r + x - 1)!}{x!(r-1)!} * p^r * q^x \), r represents the number of successes desired, x is the number of failures observed before achieving those r successes, p is the probability of success on each trial, and q is the probability of failure (where \( q = 1 - p \)). This function effectively demonstrates how the outcome of interest (the number of failures before the rth success) follows a negative binomial distribution. The factorial term provides the necessary combinatorial aspect to account for the different ways the successes and failures can occur leading up to the desired number of successes. The term \( p^r \) reflects the probability associated with the successes, and \( q^x \) accounts for the failures. The other options represent different probability distributions: one relates to the geometric distribution, another is the Poisson distribution, and another captures a basic form of a binomial function

The probability density function for the negative binomial distribution describes the probability of obtaining a specified number of successes in a series of independent Bernoulli trials, where the trials are continued until a predefined number of successes occurs. In the formula provided, ( f(x) = \frac{(r + x - 1)!}{x!(r-1)!} * p^r * q^x ), r represents the number of successes desired, x is the number of failures observed before achieving those r successes, p is the probability of success on each trial, and q is the probability of failure (where ( q = 1 - p )).

This function effectively demonstrates how the outcome of interest (the number of failures before the rth success) follows a negative binomial distribution. The factorial term provides the necessary combinatorial aspect to account for the different ways the successes and failures can occur leading up to the desired number of successes. The term ( p^r ) reflects the probability associated with the successes, and ( q^x ) accounts for the failures.

The other options represent different probability distributions: one relates to the geometric distribution, another is the Poisson distribution, and another captures a basic form of a binomial function