Different Types of probability distribution – Statanalytica

Types of probability distribution

We use probability to measure how an event is likely to happen. Numerous events that happen can’t be predicted with certainty. However, only the chance of an event to happen can be predicted. Probability ranges from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. The greater the probability of an event that has happened, the more likely it would happen. The formula of probability is the ratio of the possibility of an event to happen to the total number of outcomes. In this blog, we will focus on various Types of Probability distributions. 

What is Probability Distribution

A probability distribution is a function of mathematics that gives the probabilities of occurrence of different possible outcomes for an experiment. A table or an equation is a suitable representation of a probability distribution. This table or equation consists of every outcome of the event. 

For getting acquainted with a probability distribution, we must know about the variable and random variable.  

A variable denotes a symbol that can take any specified set of values. While a random variable is a value, a variable takes.  

Taking an example,

X depicts the random variable X.

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P(X) represents the probability of X.

P(X=x) denotes the random variable X has a specific value, denoted by X.

There are many types of probability distribution. The examples of distribution are as follows:-

Types Of Probability Distribution 

Binomial Distribution 

A binomial distribution is one of the types of probability distribution that consists of only two outcomes, namely success, and failure. If the probability of success in an event is p, then failure is 1-p. The outcomes need not be equally likely. When accounting probability, each trial is independent because the past outcome does not determine the outcome of the following toss. An experiment having only two possible outcomes repeated n number of times is called binomial. The expected value of the random variable is:- 

E(X) = 1*p + 0* (1-p) = p. 

The variance of the random variable is:-

V(X) = [E(X^2)] – [E(X)]^2. = p-p^2 = p(1-p).

Normal distributions

Normal distributions are the distributions that are used even for the most basic situations. It is one of the types of probability distribution that has the following characteristics. 

A. Mean, median, and mode coincide. 

B. The distribution curve is bell-shaped.

C. The distribution curve is symmetrical along x = μ.

D. The area under the curve is 1. 

A normal distribution in a variate X with mean μ and variance sigma^2 is a statistical distribution with probability density function. 

P(x) = 1/Sigma*sqrt(2*Pi)  *exp (-½ * ((x-μ)/(sigma))^2). 

Poisson Distribution

Poisson distribution is one of the types of probability distribution used in circumstances where cases happen at arbitrary points of space and time. Wherein we aim to know only the number of occurrences of the event. A distribution is termed as position distribution if the following assumptions are valid:-

  1. Any successful event should not affect the result of another successful event.  
  2. The probability of a successful event over a short interval should equal the probability of a successful event compared to a longer interval.
  3. The probability of a successful event in an interval approaches zero as the interval becomes smaller.
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P(k events in the interval) = exp(-events/time * time period) * (events/time*time period)^k / k! . 

Uniform Distribution

While rolling an unbiased die, the outcomes can range from 1 to 6. Hence, all the n number of possible outcomes of a uniform distribution are equally likely to happen. If we see the graph of uniform distribution, it is rectangular. That is why uniform distribution is one of the types of probability distribution called rectangular distribution. 

P(x) = 0 for x<a, 1/b-a for x belong to [a,b], 0 for x>b.

Exponential Distribution

The exponential distribution is a probability distribution that models the interval of time between the calls. It is used for the analysis of survival. It is a specific case of the gamma distribution. 

The probability density function of the above distribution, that is, an exponential distribution, is as follows:-

f(x,lambda) = lambda*exp (-lambda * x). For x greater or equal to 0. 

0 if x is less than zero.

Here lambda > 0 is the parameter of the exponential distribution, often called the rate parameter. The distribution is supported on the interval [0, ∞). If a random variable X has the distribution mentioned above, we write X ~ Exp(lambda).

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The exponential distribution exhibits infinite divisibility.

Bernoulli Distribution

Bernoulli distribution is one of the types of probability distribution in which only two outcomes are possible, that is 0 (failure) and 1 (success), and a single trial. Hence X, the random variable having a Bernoulli distribution, can take value one with the probability of success p, and the value zeroes with the probability of failure q or 1-p.

f(k,p) = if k=1, p; 1-p k=0.

Conclusion

Probability Distributions are useful in many sectors, namely, insurance, physics, engineering, computer science, and even social science, wherein the students of psychology and medicine are widely using types of probability distributions. 

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FAQ’s (Frequently Asked Questions)

What is a probability distribution used for?

Probability distributions help in modeling our world by allowing us to estimate the probability of a specific event occurring or the variability of occurrence. They’re a frequent means of describing and possibly predicting an event’s probability.

How do we use probability distributions to make decisions?

Strategy appraisals can be created using probability distributions. A scenario analysis use probability distributions to generate several theoretically unique outcomes for a given course of action or future occurrence.