Geometric Distribution

Waiting for the First Success

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The geometric distribution models the number of independent trials needed to get the first success. It is useful when each trial has only two outcomes, such as success or failure, and the probability of success stays constant from trial to trial. This makes it a natural model for coin flips, quality checks, and repeated attempts at a task. It helps answer questions about waiting time in simple random processes.

If the probability of success on each trial is $p$ , then the probability that the first success happens on trial $x$ is $P(X = x) = (1 - p)^{(x - 1)}p$ for $x = 1, 2, 3, \ldots$ . The factor $(1 - p)^{(x - 1)}$ represents failing on all previous trials, and the final $p$ represents succeeding on trial $x$ . The mean number of trials is $1/p$ , so rarer successes lead to longer expected waiting times. A special feature of the geometric distribution is its memoryless property, which means the future waiting time does not depend on how many failures have already occurred.

Key Facts

A geometric random variable X counts the trial number of the first success.
Probability mass function: $P(X = x) = (1 - p)^{(x - 1)}p$ for $x = 1, 2, 3, \ldots$
The support is x = 1, 2, 3, ... because the first success cannot occur before trial 1.
Mean: E(X) = 1/p
Variance: $\text{Var}(X) = \frac{1 - p}{p^2}$
Memoryless property: $P(X > s + t \mid X > s) = P(X > t)$

Vocabulary

Geometric distribution: A probability distribution that gives the number of trials needed to get the first success in repeated independent trials.
Independent trials: Trials are independent when the outcome of one trial does not affect the outcome of any other trial.
Success probability: The value p is the probability that a single trial results in success.
Probability mass function: A formula that gives the probability that a discrete random variable takes a specific value.
Memoryless property: A property meaning that past failures do not change the probability distribution of the remaining waiting time.

Common Mistakes to Avoid

Using x = 0 as a possible value, which is wrong because this version of the geometric distribution counts trials until the first success, so the smallest value is 1.
Forgetting the final success factor $p$ , which is wrong because $P(X = x)$ must include both $x - 1$ failures and then one success.
Applying the model when p changes from trial to trial, which is wrong because the geometric distribution requires a constant success probability on every trial.
Confusing the geometric distribution with the binomial distribution, which is wrong because geometric counts how long until the first success, while binomial counts how many successes occur in a fixed number of trials.

Practice Questions

1 A biased coin has probability of heads p = 0.3 on each flip. What is the probability that the first head occurs on the 4th flip?
2 A machine produces a defective item with probability 0.08 on each inspection, independently. Let X be the trial number of the first defective item found. Find E(X) and Var(X).
3 Explain why repeated rolls of a fair die can be modeled by a geometric distribution if success is defined as rolling a 6, and state what assumption would fail if the die were changed after each roll.

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