Question

Name and discuss four probability distributions , highlight their characteristics , how they are similar to one another , and how they differ from one another , and hoe they differ from one another?

Answer 1

The most common type of distributions are :

Normal distribution represents the behavior of most common situations in the universe . The large sum of (small) random variables often turns out to be normally distributed, contributing to its widespread application. Any distribution is known as Normal distribution if it has the following characteristics:

Binomial Distribution

A distribution where only two outcomes are possible, such as success or failure, gain or loss, win or lose and where the probability of success and failure is same for all the trials is called a Binomial Distribution.

A normal distribution is highly different from Binomial Distribution. However, if the number of trials approaches infinity then the shapes will be quite similar.

Poisson distribution

A distribution qualifies as Poisson distribution if the following assumptions are valid

1. Any successful event should not influence the outcome of another successful event.
2. The probability of success over a short interval must equal the probability of success over a longer interval.
3. The probability of success in an interval approaches zero as the interval becomes smaller.

Poisson Distribution can be called a limiting case of binomial distribution under the following conditions:

1. The number of trials is quite large .
2. The probability of success for each trial is same and indefinitely small or p →0.
3. Th mean is finite

Exponential Distribution

The exponential distribution is a continuous probability distribution which is used to model the time we need to wait before a given event occurs

It follow the below PDF

f(x) = { λe^-λx, x ≥ 0