Geometric Distribution Calculator

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Parameters

Probability of success (p):

Calculation Type:

Number of trials (x):

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Geometric Distribution: Definition, Formula, and Examples

Geometric Distribution

Definition: The geometric distribution 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.

Formula:

P(X = k) = p(1-p)^{k-1}

Where:

$k$ is the number of trials until first success
$p$ is the probability of success on each trial

Properties

Mean: $E(X) = \frac{1} {p}$
Variance: $Var(X) = \frac{1 - p} {p ^ 2}$
Support: $k = 1, 2, 3, ...$

Applications

1. Quality Control

The geometric distribution is often used in quality control to model the number of items inspected until the first defective product is discovered. This is especially useful in manufacturing processes where defects are relatively rare, and the aim is to predict how many units can be produced or tested before a defect is encountered.

2. Marketing and Sales

In marketing, the geometric distribution can be applied to model the number of customer interactions (such as calls, emails, or advertisements) required before making the first sale or acquiring the first customer. This helps businesses optimize their marketing strategies by estimating how many attempts are needed to convert a potential lead into a buyer.

3. Sports Analytics

The geometric distribution is useful in sports analytics for modeling the number of attempts needed before scoring the first point in a game. This can apply to scenarios like the number of shots a basketball player takes before scoring or how many swings a baseball player makes before hitting the ball. It helps teams and coaches analyze player performance and make strategic decisions.

4. Customer Service

In customer service operations, the geometric distribution can model the number of interactions with a customer (such as support tickets or phone calls) before resolving the customer's issue. Businesses can optimize their support processes and allocate resources more efficiently by predicting how many interactions are needed to achieve a successful resolution.

5. Network Reliability

In computer networks and communications systems, the geometric distribution models the number of transmission attempts required to successfully send data without errors. When packet loss or transmission errors occur, this distribution helps estimate how many retries will be needed to achieve a successful transfer, assisting in network design and reliability analysis.