## Geometric Distribution

### Defining Geometric Distribution

The Geometric Distribution defines the count of trials needed to secure the first successful outcome in a series of independent Bernoulli trials. Each trial holds a fixed probability of success, represented by p. The probability mass function (PMF) for a Geometric Distribution follows the equation:

P (X = k) = (1-p) ^ (k-1) * p

In this equation, X embodies the random variable representing the count of trials, k stands for the count of trials required for the initial successful outcome, and p denotes the probability of success in a single trial.

### Properties of Geometric Distribution

Several distinctive properties characterize the Geometric Distribution:
Memorylessness: This distribution demonstrates memorylessness, meaning that the likelihood of success in future trials remains independent of the result of past trials.
Expectation and Variance: The expected value or mean of a Geometric Distribution equals 1/p, and the variance equates to (1-p) / p^2.
Skewness: The Geometric Distribution tends towards positive skewness, with the skewness rising in proportion to a decrease in the probability of success.

### Applications of Geometric Distribution

The Geometric Distribution serves numerous purposes across various fields, comprising:
Reliability Engineering: The distribution contributes to modeling the count of trials necessary before a system failure takes place.
Quality Control: In manufacturing contexts, the Geometric Distribution can depict the count of items inspected before finding a defective one.
Biology: This distribution facilitates modeling the count of trials needed for a particular genetic event to appear in a population.

### Geometric Distribution in Statistical Analysis

The Geometric Distribution is commonly leveraged in statistical analyses to discern the likelihood of success and the predicted count of trials for a specific event. Moreover, it is employed in hypothesis testing to compare the observed count of trials with the count anticipated under a declared probability of success.
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