Do you know about every Probability Distribution function with cool examples?

 

Probability Distributions: It is a mathematical formula which helps us to find out the chances of occurring of an event. If we say that our dataset belongs to some particular distribution then we can say that probability of events in such dataset behave in a particular pattern. Say, the dataset is from uniform distribution then each and every event of range will have equal chances of occurring.

 

Types of distributions:

Discrete Distributions: Distributions defined for discrete random variables (which can take countable values such as 0,1,2,3….) are known as discrete distributions. Following given distributions are discrete:

Uniform Distribution: Data is said to be uniformly distributed if the probability of events is equal.                                              

For example: Events in throw of a dice or flipping a coin have equal probabilities. The number of bouquets sold daily at a flower shop is uniformly distributed with a maximum of 40 and a minimum of 30.

Bernoulli Distribution: When in a single random trial only two outcomes are possible and let the trial has a probability of success ‘p’ and probability of failure ‘1-p’ then this trial is termed as Bernoulli trial and corresponds to Bernoulli distribution.

For example:  There are 5 black and 4 white balls in a bag. A single ball is drawn from it and the probability of drawing a white ball is 4/9. This is an example of Bernoulli as only two outcomes are possible i.e. either white or black ball can be drawn and only one ball is drawn hence single trial is being conducted.


 


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Binomial Distribution : When in any particular experiment only two outcomes are possible  (e.g. pass/fail, smoker/non-smoker  etc.)where one outcome is termed as success with probability ‘p’ and other as failure with probability ‘1-p’ and “n” independent trials are done for this experiment then it is said to follow a binomial distribution with parameters n and p .

 For example: Results of IAI are declared as either pass or fail, no grading or marks are declared which means there are only two possible outcomes either pass or fail. Say the probability of passing a student is ‘p’ and number of trials are number of students appeared.

[When n=1 binomial distribution is known as Bernoulli distribution.]

 

Poisson Distribution: The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant rate and independently of the time. This distribution models the rate, average, pattern or arrival path of various events.

Limiting Case: Poisson distribution can be described as a limiting case of binomial distribution when n is sufficiently large and p is sufficiently small. Hence, in binomial distribution when n tends to infinity and p tends to 0 the binomial distribution can be approximated to Poisson distribution having ฮป=np. In general cases this approximation is used when n>100 and p<0.05.

For example: The number of claims arising on an insurance policy are 5 per year on an average and if happening of a claim doesn’t affect the other future claims then it follows Poisson distribution. Some other examples may be the number of accidents occurring per day in a city, number of mails received in a day, number of patients arriving in an emergency room between 10 and 11 pm.

 

Geometric distribution: In case of geometric distribution we will keep doing trials until we get our first success where number of trials can be infinite. This is the only discrete distribution which follows memoryless property which states that the failures on first n trials does not affect the probability of success on upcoming trials.

For example: In case of a life insurance policy a person will give premiums until event of death. Here, death will be our success with say probability ‘p’.

[The above given description is of Type 1 Geometric. In case of Type 2 Geometric we consider number of failures before first success.]

 

Negative Binomial Distribution: In case of negative binomial distribution we will keep    doing trials until we get our Kth success. Here, we find the probability of having kth success in xth trial. If we put K=1 then this will become geometric distribution. In fact, it can be expressed as a sum of K geometric variables.

For example: In case of acceptance sampling plans a batch of say 60 items is inspected until defectives, if we get 10 defectives then the batch is rejected otherwise accepted. The probability of a batch being rejected on finding 10 defectives in less than 60 trials can be calculated using negative binomial distribution.

 [The above given description is of Type 1 negative binomial and it counts the number of trials up to and including the kth success whereas Type 2 counts the failures before the kth success. So, for the above example if we found 10th defective on say 30th trial then in Type 1 X will be 30 as we had to do 30 trials before 10th success but in case of Type 2 X will be 20 because we had 20 failures before finding 10th success.]

 

 

Hyper Geometric Distribution: This distribution helps to find the probability the probability of k successes in n random draws from a population of size N.

For example: We want to conduct a survey about drug usage and we randomly select 20 students among 5000 boys and 4300 girls of a university. To find the probability that out of these randomly chosen 20 students 8 will be girls hypergeometric will be helpful.

Example for Discrete Distributions:  

Let there are 50 people out of which on 18 are graduated and rest are not.

--Let the probability of a person being graduated is 0.6 and is known in advance then the probability of selecting two people who are graduated will be modeled using binomial distribution.

--Now, if we randomly select people from this group and keep selecting until we select a person who is graduated can be modeled using geometric distribution and the probability of selecting 1st graduate on 6th trial can be known using geometric distribution.

--Now, if we want to know the probability of selecting 3rd graduate in 10th trial then this can be modeled using Negative binomial distribution.

--Now, if we randomly select 7 people from the group and the probability of having 2 graduates among these 7 can be found using hypergeometric distribution.

 

Continuous Distributions: Distributions defined for continuous random variables (which can take uncountable infinite number of values) are known as continuous distributions. Following given distributions are continuous:

Uniform Distribution:  When a continuous random variable is uniformly distributed in an interval such that the probability of occurrence is equally distributed then it is said to follow uniform distribution.

For example: If at a metro station trains run every five minutes. Then the waiting time for a person entering the station at a random time will follow uniform (0,5).

[Uniform distributions are also widely used for random number simulations because cdf of every distribution follows U(0,1).]

Gamma Distribution: Gamma distribution is a positively skewed continuous distribution having two parameters ฮฑ and ฮฒ where ฮฑ is the shape parameter and ฮฒ is the scale parameter. It can also be described as ฯ‡2 distribution with m degrees of freedom when ฮฑ=m/2 and ฮฒ=1/2.

For example: Gamma distributions are used to model the size of insurance claims or the size of loan defaults.

[Gamma distributions are also widely used as conjugate prior for various distributions in Bayesian statistics.]

 

Exponential Distribution: It is the probability distribution of time between events in a Poisson process. The summation of K exponential random variables follows gamma with K and ฮป where ฮป is parameter of exponential distribution.

For example: If we note the time between arrival of students in a classroom or patients in a hospital. Then the data of these time intervals will follow a continuous distribution.

[Exponential distribution also follows memoryless property.]

Beta Distribution: Beta is a positive continuous distribution defined on an interval of (0,1) having two positive parameters ฮฑ and ฮฒ. It is widely used in Bayesian statistics as a conjugate prior for various distributions.

 

T distribution: This distribution is similar to normal distribution in terms of symmetry and shape but has heavier tails and hence produces values far from mean. It is used in parameter estimation of ยต when sample size is small and ฯƒ2 is unknown.  

ฯ‡2 Distribution: ฯ‡2 distribution with n degrees of freedom is a sum of n squared standard normal variates. It is widely used in testing of population variances and chi-squared tests like goodness of fit and contingency tables

F Distribution: F distribution with (m-1, n-1) degrees of freedom is ratio of ฯ‡2 with m degree of freedom and ฯ‡2 with n degrees of freedom. It is used in hypothesis testing of ratio of population variances.

Weibull Distribution: Weibull is a probability distribution which is widely used in data analysis having two parameters ฮฑ and ฮณ where c is shape parameter and ฮณ is scale parameter. The value for the shape parameter (ฮณ) determines the hazard rates:

--If gamma is less than 1, then the hazard rate decreases with time (i.e. the process has a large number of infantile or early-life failures and fewer failures as time passes).

--For ฮณ = 1: the hazard rate is constant, which means it’s indicative of useful life or random failures.

--If ฮณ > 1: the hazard rate increases with time (i.e. the distribution models wear-out failures, which tend to happen after some time has passed).

For example: Weibull analysis can be used to study survival models, lifetimes of medical and dental implants, Components produced in a factory, Warranty analysis, Utility services etc.

 

Pareto Distribution: Pareto distribution was firstly discovered to model the distribution of income having two parameters ฮฑ and ฮป where ฮฑ is the shape parameter and ฮป is lower bound on data.

For example: Pareto distributions can be widely used to model the distribution of incomes, city populations, lifetime of manufactured items, claim amounts of insurance policies.

 ๐’๐ฎ๐›๐ฌ๐œ๐ซ๐ข๐›๐ž ๐ญ๐จ ๐ฆ๐ฒ ๐˜๐จ๐ฎ๐“๐ฎ๐›๐ž ๐‚๐ก๐š๐ง๐ง๐ž๐ฅ ๐ญ๐จ ๐ฅ๐ž๐š๐ซ๐ง ๐๐ฒ๐ญ๐ก๐จ๐ง ๐š๐ง๐ ๐’๐๐‹ ๐Ÿ๐จ๐ซ ๐€๐œ๐ญ๐ฎ๐š๐ซ๐ข๐ž๐ฌ

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Normal Distribution: Normal distribution is a symmetrical and bell-shaped distribution with mean ยต and variance ฯƒ2. According to central limit theorem, all the distributions tends to normal for large sample size. In case of normal distribution mean = median =mode and ยต represents the middle point of the graph while ฯƒ2 represents the spread. It plays a significant role in statistics as it is widely used for approximating various distributions, hypothesis testing, regression models etc.

 

Standard Normal Distribution: A variable Z is said to follow standard normal distribution with ยต=0 and ฯƒ2 = 1 when Z=(x-ยต)/ฯƒ. Normal variates are transformed into Z variates (i.e. standard normal variates) to find P(Z<=z) as we have table available for only Z variables.

 

Lognormal Distribution: Lognormal distribution is a positively skewed continuous distribution having 2 parameters ยต and ฯƒ2 which are not it’s mean and variance. If X follows a lognormal distribution, then Y= log X has a normal distribution with mean =ยตand variance=ฯƒ2.

      

Burr Distribution: The Burr distribution is a unimodal family of distributions with a wide variety of shapes. This distribution is used widely to model crop prices, household income, option market price distributions, risk (insurance) and travel time.

 

๐’๐ฎ๐›๐ฌ๐œ๐ซ๐ข๐›๐ž ๐ญ๐จ ๐ฆ๐ฒ ๐˜๐จ๐ฎ๐“๐ฎ๐›๐ž ๐‚๐ก๐š๐ง๐ง๐ž๐ฅ ๐ญ๐จ ๐ฅ๐ž๐š๐ซ๐ง ๐๐ฒ๐ญ๐ก๐จ๐ง ๐š๐ง๐ ๐’๐๐‹ ๐Ÿ๐จ๐ซ ๐€๐œ๐ญ๐ฎ๐š๐ซ๐ข๐ž๐ฌ

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