Geometric distribution

Definition

The geometric distribution is a discrete probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. Each trial has only two possible outcomes: success or failure, and the probability of success remains constant across trials. This distribution is key for understanding scenarios where we are interested in the number of attempts required until an event occurs, making it a fundamental concept in discrete probability.

5 Must Know Facts For Your Next Test

1. In a geometric distribution, the probability of success on each trial is denoted as 'p', while the probability of failure is 'q = 1 - p'.
2. The expected value (mean) of a geometric distribution is calculated using the formula $$E(X) = \frac{1}{p}$$, indicating how many trials are expected before the first success.
3. The variance of a geometric distribution can be computed using $$Var(X) = \frac{q}{p^2}$$, showing how much the number of trials varies from the expected value.
4. Geometric distributions are memoryless, meaning the probability of success in future trials does not depend on past failures.
5. The geometric distribution is often used in scenarios such as finding out how many coin flips are needed until we get heads for the first time.

Review Questions

• How does the geometric distribution relate to Bernoulli trials, and why is this connection important?
  • The geometric distribution is based on a sequence of Bernoulli trials, where each trial has two outcomes: success or failure. This relationship is crucial because it allows us to model real-world scenarios where we want to know how many trials are required to achieve the first success. Understanding this connection helps us apply the geometric distribution effectively in various situations like quality control or reliability testing.
• Calculate the expected number of trials needed to get the first success if the probability of success on each trial is 0.2.
  • To calculate the expected number of trials needed for the first success in a geometric distribution, we use the formula $$E(X) = \frac{1}{p}$$. In this case, with p = 0.2, the expected number of trials would be $$E(X) = \frac{1}{0.2} = 5$$. This means that, on average, it will take 5 trials to achieve one success.
• Evaluate how the memoryless property of geometric distribution influences decision-making in scenarios involving repeated trials.
  • The memoryless property of the geometric distribution means that past failures do not affect future probabilities; every trial is independent. This influences decision-making significantly because it simplifies predictions. For example, if you flip a coin and get tails multiple times in a row, the probability of getting heads on the next flip remains constant at 50%. This understanding helps in strategic planning and risk assessment when dealing with repeated random experiments.

Related terms

Bernoulli trial: A Bernoulli trial is a random experiment with exactly two possible outcomes: success or failure, usually represented as 1 and 0 respectively.

Probability mass function (PMF): The PMF of a discrete random variable gives the probability that the variable takes on a specific value, showing how probabilities are distributed across different values.

Negative binomial distribution: A negative binomial distribution generalizes the geometric distribution by modeling the number of trials needed to achieve a specified number of successes, rather than just one.