# [[Geometric Distribution]] (Mathematics > Statistics) ## Definition The **geometric distribution** is a discrete probability distribution that models the number of independent Bernoulli trials needed to achieve the first success. If each trial has a probability $p$ of success, then the probability that the first success occurs on the $k$-th trial is given by: $ P(X = k) = (1-p)^{k-1}p, \quad k = 1, 2, 3, \ldots $ ## Key Concepts - **Bernoulli Trials:** Each trial has exactly two outcomes: success or failure. - **Waiting Time:** It measures the number of trials until the first success. - **Memoryless Property:** The probability of success in future trials is independent of the past. - **Discrete Distribution:** The variable $X$ takes on positive integer values. ## Important Properties 1. **Memorylessness:** The geometric distribution is the only discrete distribution with the memoryless property. 2. **Mean:** The expected number of trials until the first success is $\frac{1}{p}$. 3. **Variance:** The variance is given by $\frac{1-p}{p^2}$. ## Essential Formulas - **Probability Mass Function:** $ P(X = k) = (1-p)^{k-1}p. $ - **Mean and Variance:** $ E(X) = \frac{1}{p}, \quad \text{Var}(X) = \frac{1-p}{p^2}. $ ## Core Examples 1. **Basic Example:** Suppose a fair die is thrown repeatedly until a six appears. Here, a "success" (getting a six) has probability $p=\frac{1}{6}$. The probability that the first six occurs on the 3rd throw is: $ P(X=3) = \left(1-\frac{1}{6}\right)^{3-1}\left(\frac{1}{6}\right) = \left(\frac{5}{6}\right)^2\left(\frac{1}{6}\right). $ 2. **Advanced Example:** Consider a scenario where a machine has a $10\%$ chance of producing a defective item (failure). The probability that the first defective item is produced on the 4th trial is: $ P(X=4) = (0.9)^{3}(0.1). $ ## Related Theorems/Rules - **Memoryless Property:** For any integers $m$ and $n$, $ P(X > m+n \mid X > m) = P(X > n). $ - **Negative Binomial Distribution:** The geometric distribution is a special case where the number of required successes is one. ## Common Pitfalls - **Indexing Error:** Remember that the count starts at $k=1$, not $0$. - **Misidentifying the Scenario:** Confusing the geometric distribution with the binomial distribution, which counts the number of successes in a fixed number of trials. - **Incorrect Exponent:** Using $(1-p)^k$ instead of $(1-p)^{k-1}$ in the PMF. ## Related Topics - [[Bernoulli Distribution]] - [[Negative Binomial Distribution]] - [[Exponential Distribution]] - [[Discrete Random Variables]] - [[Expected Value of Discrete Random Variables]] - [[Conditional Probability]] - [[Multiplication Law for Probability]] - [[Cumulative Distribution Function]] - [[Binomial Theorem]] - [[Normal Distribution]] - [[Standard Normal Distribution]] - [[Law of Total Probability]] - [[Memoryless Property]] - **Bernoulli Distribution:** The distribution of a single trial with two outcomes. - **Negative Binomial Distribution:** A generalization of the geometric distribution for the number of trials until a fixed number of successes. - **Exponential Distribution:** The continuous counterpart of the geometric distribution. ## Quick Review Questions 1. What is the probability mass function of the geometric distribution? 2. If a fair die is tossed repeatedly, what is the probability that the first six appears on the 3rd toss? *** ![[Geometric_Distribution_visualization]]