Geometric Distribution

Definition

The geometric distribution is a discrete probability distribution that models the number of independent Bernoulli trials (successes or failures) required to get the first success. It represents the probability of the first success occurring on a specific trial in a sequence of independent trials, each with a constant probability of success.

5 Must Know Facts For Your Next Test

1. The geometric distribution is commonly used to model the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials.
2. The parameter of the geometric distribution is the probability of success on a single trial, denoted as $p$.
3. The probability mass function (PMF) of the geometric distribution is given by $P(X = x) = p(1-p)^{x-1}$, where $x$ represents the number of trials.
4. The expected value (mean) of the geometric distribution is $E[X] = \frac{1}{p}$, and the variance is $Var(X) = \frac{1-p}{p^2}$.
5. The geometric distribution has the memoryless property, meaning that the probability of success on any given trial is independent of the outcomes of previous trials.

Review Questions

• Explain the relationship between the geometric distribution and Bernoulli trials.
  • The geometric distribution models the number of independent Bernoulli trials required to obtain the first success. In a Bernoulli trial, there are two possible outcomes: success or failure, with a constant probability of success on each trial. The geometric distribution represents the probability of the first success occurring on a specific trial in a sequence of these independent Bernoulli trials.
• Describe the memoryless property of the geometric distribution and its implications.
  • The geometric distribution has the memoryless property, which means that the probability of success on any given trial is independent of the outcomes of previous trials. This implies that the probability of obtaining the first success on a particular trial is the same, regardless of the number of trials that have already occurred without a success. This property allows the geometric distribution to be used to model a wide range of real-world phenomena where the probability of success remains constant across trials.
• Analyze the relationship between the parameters of the geometric distribution and its expected value and variance.
  • The geometric distribution has a single parameter, $p$, which represents the probability of success on a single trial. The expected value (mean) of the geometric distribution is $E[X] = \frac{1}{p}$, meaning that the average number of trials required to obtain the first success is inversely proportional to the probability of success on a single trial. The variance of the geometric distribution is $Var(X) = \frac{1-p}{p^2}$, which indicates that as the probability of success $p$ increases, the variance decreases, resulting in a more predictable number of trials needed to obtain the first success.

Related terms

Bernoulli Trial: A Bernoulli trial is a random experiment with two possible outcomes, usually labeled as 'success' and 'failure,' where the probability of success remains constant across all trials.

Probability Mass Function (PMF): The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value.

Memoryless Property: The memoryless property of the geometric distribution means that the probability of success on any given trial is independent of the outcomes of previous trials.