About Geometric Distribution Formula
In a Bernoulli trial, a geometric distribution is a sort of discrete probability distribution that shows the likelihood of a number of sequential failures before a success. A Bernoulli trial is a scientific experiment with only two possible outcomes either success or failure, or In other words, a Bernoulli trial is repeated until success is attained and then stopped in geometric distribution.
In various real-life circumstances, the geometric distribution is commonly used. In the financial industry, for example, the geometric distribution is used to estimate the financial rewards of a certain option in a cost-benefit analysis.
What do you mean by Geometric Distribution?
Geometric distributions are probability distributions that are based on three assumptions. Following is a list of them.
- Trials are being undertaken independently.
- Each trial can only have two outcomes: success or failure.
- For each trial, the success probability, represented by p, is the same.
Geometric Distribution Definition
A geometric distribution is a discrete probability distribution that depicts the likelihood of achieving success following a series of failures. Until the first success, a geometric distribution can have an infinite number of tries.
Geometric Distribution Example
Assume you keep rolling the dice until you get a "3." The probability of finding "3" is then p = 1/6, and the random variable, X, can take on any value between 1 and 6, until the first success is achieved. With p = 1 / 6, this is an example of a geometric distribution.
Formula for Geometric Distribution
- P(X = x) = (1 - p)x - 1p
- P(X ≤ x) = 1 - (1 - p)x
Probability mass function (pmf) and cumulative distribution function can both be used to characterise a geometric distribution (CDF). The likelihood of a trial's success is denoted by p, whereas the probability of failure is denoted by q. Here, q = 1 - p. The geometric probability distribution of a discrete random variable, X, is represented as X ∼G (p). The formulas for the pmf and CDF of a geometric distribution are given below.
Geometric Distribution PMF
Likelihood that a discrete random variable, X, will be exactly identical to some value, x, is determined by probability mass function. The following is the formula for geometric distribution pmf:
- P(X = x) = (1 - p)x - 1 p, Here, 0 < p ≤ 1.
Geometric Distribution CDF
The probability that a random variable, X, will assume a value less than or equal to x can be described as the cumulative distribution function of a random variable, X, that is assessed at a point, x. Distribution function is another name for it. The following is the geometric distribution CDF formula:
- P(X ≤ x) = 1 - (1 - p)x
Mean of Geometric Distribution
Geometric distribution's mean is also geometric distribution's expected value. Weighted average of all values of a random variable, X, is expected value of X. The mean of a geometric distribution can be calculated using the following formula:
- E[X] = 1 / p
The variance of Geometric Distribution
Variance is the measure of dispersion that examines how far data in distribution is spread out in relation to the mean. The variance of a geometric distribution is calculated as follows:
- Var[X] = (1 - p) / p2
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