What represents the probability density function for the geometric distribution?

What represents the probability density function for the geometric distribution?

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

Answer: (A)

The geometric distribution models the number of trials until the first success in a series of Bernoulli trials, where each trial has a success probability of \( p \) and a failure probability of \( q = 1 - p \). The probability density function (PDF) for the geometric distribution is defined as the probability that the first success occurs on the \( x \)-th trial. This is given by: \[ f(x) = q^{x-1} * p \] Here, \( q^{x-1} \) represents the probability of failing the first \( x-1 \) trials, and \( p \) is the probability of succeeding on the \( x \)-th trial. It is essential to note that \( x \) can take values 1, 2, 3, ..., which reflects that the first success can occur at any positive integer trial. In the correct representation of the function, \( q \) is raised to the power of \( x-1 \) instead of \( x \) to account for the \( x \)-th success occurring after \( x-1 \) failures. The option chosen accurately portrays the needed structure of success and failures in the context of the geometric distribution for trial

The geometric distribution models the number of trials until the first success in a series of Bernoulli trials, where each trial has a success probability of ( p ) and a failure probability of ( q = 1 - p ).

The probability density function (PDF) for the geometric distribution is defined as the probability that the first success occurs on the ( x )-th trial. This is given by:

[ f(x) = q^{x-1} * p ]

Here, ( q^{x-1} ) represents the probability of failing the first ( x-1 ) trials, and ( p ) is the probability of succeeding on the ( x )-th trial. It is essential to note that ( x ) can take values 1, 2, 3, ..., which reflects that the first success can occur at any positive integer trial.

In the correct representation of the function, ( q ) is raised to the power of ( x-1 ) instead of ( x ) to account for the ( x )-th success occurring after ( x-1 ) failures. The option chosen accurately portrays the needed structure of success and failures in the context of the geometric distribution for trial