Probability is the chances of occurrence of an event. Probability deals with two types of distributions – geometric and binomial. Before coming to geometric distribution, let us know what the Bernoulli trial is.
Bernoulli trial is a way to find the probability of an event. It has only two outcomes – success and failure. The geometric distribution is a way to find the probability of these events if they occur repeatedly.
In simpler terms, the geometric distribution represents the probability of an event when success is obtained in a Bernoulli trial. The trials are repeated infinitely till success comes, even though failure occurs repeatedly. The trial ends once a success occurs.
For example, if we need to find the probability of 3 in a dice occurrence, we have to roll the dice till 3 comes. The trial will end if 3 comes in one trial or after 200 trials.
The geometric distribution is mostly used in financial sectors to do cost-benefit analyses, estimate cost and income benefits, and make a certain financial decision based on the studies and calculations after applying geometric distributions.
There are three assumptions that need to keep in mind while dealing with geometric distributions:
1) The trials conducted in this distribution are independent.
2) There can only be two outcomes of every event – success and failure.
3) The probability of success is denoted by p, and a failure by q, in most of the texts.
P (X = x) = (1 – p)⁻¹ p
P (X ≤ x) = 1 – (1 – p)ˣ
Where, p = probability of success
X = discrete random variable
x = a value where analysis is to done
The first formula is derived from the probability mass function (PMF). And the second formula is derived from the cumulative distribution function (CDF).
The mean value of a geometric distribution is known as the expected value of the geometric distribution. The expected value is represented by X, which is a weighted average of all values of X. The mean is denoted by:
E [X] = 1/p
The variance in a geometric distribution checks how far the data is spread out with respect to the mean within the distribution. The formula to derive a variance is:
Var [X] = (1 – p) / p²
The root of variance is known as the standard deviation. The formula of standard deviation is:
Geometric Distribution | Binomial Distribution |
A geometric distribution deals with the first success only. The random variable, X, denotes the number of trials that occur to obtain that first success. | In a binomial distribution, there are a fixed number of trials. The random variable, X, counts the number of successes that occur in those trials. |
The probability mass function is given by PMF = (1 - p)ˣ⁻¹ p | The probability mass function is given by PMF = pˣ (1 − p)ⁿ⁻ˣ |
Mean = 1 / p, Variance = (1 - p) / p² | Mean = np, Variance = np (1 - p) |