What is the probability mass function of the geometric distribution?

What is the probability mass function of the geometric distribution?

• A. F(x) = 1 - q^(x+1)
• B. F(x) = q^x / p
• C. F(x) = p^x
• D. F(x) = qr^x

Answer: (A)

The probability mass function of the geometric distribution pertains to the number of trials required to achieve the first success in a sequence of independent Bernoulli trials, where each trial results in success with probability \( p \) and failure with probability \( q = 1 - p \). The probability that the first success occurs on the \( x \)-th trial can be expressed as follows: 1. For the first \( x-1 \) trials, there must be \( x-1 \) failures, each with probability \( q \). 2. The \( x \)-th trial must result in success, which occurs with probability \( p \). Thus, the probability that the first success occurs on the \( x \)-th trial is given by: \[ P(X = x) = q^{x-1} \cdot p \] This formula reflects that the first \( x-1 \) trials are failures, followed by a success at the \( x \)-th trial. The cumulative distribution function, reflecting the probability that the first success occurs on or before the \( x \)-th trial, can be derived from the probability mass function. It is: \[ F(x) = P(X \leq x) =

The probability mass function of the geometric distribution pertains to the number of trials required to achieve the first success in a sequence of independent Bernoulli trials, where each trial results in success with probability ( p ) and failure with probability ( q = 1 - p ).

The probability that the first success occurs on the ( x )-th trial can be expressed as follows:

1. For the first ( x-1 ) trials, there must be ( x-1 ) failures, each with probability ( q ).

2. The ( x )-th trial must result in success, which occurs with probability ( p ).

Thus, the probability that the first success occurs on the ( x )-th trial is given by:

[

P(X = x) = q^{x-1} \cdot p

]

This formula reflects that the first ( x-1 ) trials are failures, followed by a success at the ( x )-th trial.

The cumulative distribution function, reflecting the probability that the first success occurs on or before the ( x )-th trial, can be derived from the probability mass function. It is:

[

F(x) = P(X \leq x) =