Geometric distribution

Definition

Geometric distribution is a probability distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, where each trial has the same probability of success. This distribution is crucial in understanding scenarios where we want to know how many attempts it takes before we succeed, such as flipping a coin until it lands on heads.

5 Must Know Facts For Your Next Test

1. The probability of getting the first success on the k-th trial is given by the formula $$P(X = k) = (1-p)^{k-1} p$$, where p is the probability of success.
2. The geometric distribution is memoryless, meaning that the probability of success on future trials does not depend on past trials.
3. The expected value (mean) of a geometric distribution is given by $$E(X) = \frac{1}{p}$$.
4. The variance of a geometric distribution can be calculated using the formula $$Var(X) = \frac{1-p}{p^2}$$.
5. Geometric distribution applies to scenarios such as waiting for the first customer to make a purchase or rolling a die until a six appears.

Review Questions

• Explain how the memoryless property of geometric distribution affects its applications in real-world situations.
  • The memoryless property means that past trials do not affect future outcomes, so each trial is independent. This property is essential in real-world applications like quality control, where each product's defect status does not depend on previous products. It simplifies calculations because it allows us to treat each trial as a new event without needing to consider historical data.
• How can you derive the expected value and variance formulas for a geometric distribution using its probability mass function?
  • To derive the expected value $$E(X)$$ for geometric distribution, we sum over all possible values: $$E(X) = \sum_{k=1}^{\infty} k P(X=k) = \sum_{k=1}^{\infty} k (1-p)^{k-1} p$$. Using calculus and series manipulation, this results in $$E(X) = \frac{1}{p}$$. For variance, we first find $$E(X^2)$$ using similar methods and then apply the formula $$Var(X) = E(X^2) - (E(X))^2$$. This gives us $$Var(X) = \frac{1-p}{p^2}$$.
• Evaluate the implications of using geometric distribution in modeling scenarios with high success probabilities versus low success probabilities.
  • When modeling scenarios with high success probabilities, the number of trials needed to achieve success tends to be low, resulting in smaller expected values and variance. Conversely, in situations with low success probabilities, achieving the first success may require significantly more trials, leading to higher expected values and variances. This difference can greatly impact decision-making processes, resource allocation, and risk assessment in fields like marketing and quality assurance, where understanding how often successes occur directly informs strategy.

Related terms

Bernoulli trial: A random experiment with exactly two possible outcomes: success or failure.

Success:

The desired outcome in a probability experiment, often defined as achieving a specific result.

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