Soji Sam

What is probability distribution -
A probability distribution is a statistical function that specifies all potential values and probabilities for a random variable within a certain range.
 
2 Types of Probability Distribution:
Distributions of Discrete Probability : A discrete distribution indicates the likelihood of each value of a discrete random variable occurring. These distributions frequently include statistical assessments of "counts" or "the number of times" an event happens. For example, discrete events, such as tossing dice or flipping coins, have a finite number of outcomes.
 
Distributions of Continuous Probability: A continuous distribution explains the probability of the various values of a continuous random variable. A continuous random variable can have an unlimited and uncountable number of values (known as the range). Time, for example, is infinite: you might count from 0 seconds to a million seconds...a quadrillion second...and so on infinitely.
 
 
Normal distribution -
The normal distribution is perhaps the most typical probability distribution and it is continuous distribution. The data tends to cluster around a centre value with no bias to the left or right, and it approaches a "Normal Distribution" in the following way:
 

The normal distribution has
50% of values are less than the mean, whereas 50% are more than the mean There is a symmetry in the centre Mean, median and mode are equal  
Few examples are:
Heights of men - 
 
Rolling 2 dices -  
 
 
Performance - 
 
 
There are other distributions which have a shape that resembles a bell curve or a Normal distribution. Let's look at few of them -
 
Student's T Distribution -
The Student's T Distribution is a family of distributions that resemble the normal distribution curve but are somewhat shorter and thicker. The Student's T Distribution (and the accompanying t scores) are used in hypothesis testing to determine whether the null hypothesis should be accepted or rejected. It gives the center a lower probability and the tails a larger probability than the normal distribution.
 
When to use Student's T distribution:
When there are few samples, the student's t distribution is utilized instead of the normal distribution. This distribution resembles the normal distribution more as the sample size increases. Indeed, for sample sizes greater than 20 (i.e. more degrees of freedom), the distribution closely resembles the normal distribution. The standard deviation of the population is unknown. The distribution of the population is skewed.
 
 
Example: Measure the average test score from a sample of just 20 students.
The Student's T-distribution should be used to determine the confidence interval around the mean. Your confidence interval will be artificially precise if you utilize the normal distribution (z-distribution)
.
 
Logistic Distribution -
The logistic distribution is also a distribution of continuous probability. The shape is similar to the normal distribution, but the tails are heavier (higher kurtosis).The fundamental distinction between the normal and logistic distributions is in the tails and the behaviour of the failure rate function. The tails of the logistic distribution are somewhat longer than those of the normal distribution.
The distribution's shape is determined by two parameters:
The location parameter indicates where the x-axis is centred. The scale parameter indicates the spread. Because the logistic distribution is symmetric, the mean, median, and mode are all the same.
 
When to use Logistic distribution:
The logistic distribution is primarily utilized since the cumulative distribution formula is reasonably straightforward to deal with. The formula very closely approximates the normal distribution. Looking up numbers in the z-table and rounding up or down to the closest z-score is normally how you find cumulative probabilities for the normal distribution. Because the cumulative distribution function is so complex to deal with, exact values are generally discovered using statistical software.
Although there are numerous other functions that can approach the normal, their mathematical formulations are typically exceedingly complex. In comparison, the logistic distribution has a considerably simpler CDF formula.
 
 
Example: 
The logistic distribution has been utilised in growth models and in a kind of regression called as logistic regression.  Also it uses to calculate the relative skill level of chess players.  
 
 
Binomial Distribution -
The binomial distribution is a discrete probability distribution that produces just two outcomes in an experiment: success or failure.
 
In a binomial distribution, two parameters, n and p, are used. The variable 'n' indicates the number of times the experiment is repeated, and the variable 'p' indicates the likelihood of any given outcome.
 
Below mentioned are the few properties of binomial distribution-
There are 2 possible outcomes: success or failure, true or false, yes or no. There are a certain number 'n' of independent trials. For each trial, the likelihood of success or failure remains constant. Only the number of successes from n separate trials is computed.  
How it is be different from normal distribution:
The primary distinction between the binomial and normal distributions is that the binomial distribution is discrete, whereas the normal distribution is continuous. The binomial distribution has a finite number of occurrences, whereas the normal distribution has an infinite number of events. If the sample size for the binomial distribution is sufficiently big, the binomial distribution's distribution curve is comparable to the normal distribution curve.
 

 
Example: 
Flipping a coin - there are just two conceivable results if we flip a coin: either heads or tails To find the number of male and female employees in an organization