Statistical distributions

Asit

3 years ago

Types of Statistics – distributions

Bernoulli Distribution
Uniform Distribution
Binomial Distribution
Normal Distribution
Poisson Distribution
Exponential Distribution

Bernoulli distribution

A Bernoulli distribution has only two possible outcomes,
1 success and 0 failure in a single trial.
Random variable X which has a Bernoulli distribution can take the value 1 with the probability of success, say p, and the value 0 with the probability of failure, say q (q = 1-p).

The probability mass function is given by: p^x(1-p)^1-x where x € (0, 1).

Uniform Distribution

When you roll a fair die, the outcomes are 1 to 6. The probabilities of getting these outcomes are equally likely and that is the basis of a uniform distribution.
All the n number of possible outcomes of a uniform distribution are equally likely.
A variable X is said to be uniformly distributed if the density function is:

The graph of a uniform distribution curve looks like

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 the same for all the trials is called a Binomial Distribution.

Each trial is independent.
There are only two possible outcomes in a trial- either a success or a failure.
A total number of n identical trials are conducted.
The probability of success and failure is the same for all trials. (Trials are identical.)
If p is the probability of success, q is the probability of failure, The mathematical representation of binomial distribution is given by:

The Binomial distribution graph (if, Probability of success = Probability of failure)

The Binomial distribution graph (if, Probability of success ≠ Probability of failure)

Normal Distribution

The normal distribution represents the behaviour of most of the situations in the universe. Therefore it is called a “normal” distribution.
For normal distribution mean, median, and mode are common points.
The curve of the distribution is bell-shaped and symmetrical about the mean μ.
The total area under the probability curve is 1.
Exactly half of the values are to the left of the center and the other half to the right.
A normal distribution is highly different from Binomial Distribution.
if a number of trials in the Binomial Distribution approach infinity then the shapes will be quite similar to Normal distribution.
The mathematical representation of Normal distribution is given by:

Poisson distribution

Poisson distribution is applicable in situations where events occur at random points of time and space and focus on the number of occurrences of the event.
A distribution is called a Poisson distribution when the following assumptions are valid:

1. 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.

Examples:-

- - The number of printing errors per page in a book.
  - The number of thefts reported in a police station per day.
  - The number of patients treated per day in hospital OPD.
  - The number of customers visiting a shopping mall per day.
The mathematical representation of Normal distribution is given by:

Exponential Distribution

The exponential distribution is widely used for survival analysis.
Examples: Expected life of a machine, Expected life of a human.
A random variable X is said to have an exponential distribution with a Probability distribution function:

f(x) = { λe^-λx, x ≥ 0 , λ > 0 it is called the rate.

r survival analysis, λ is called the failure rate of a device at any time t, given that it has survived up to t.

- - P{X≤x} = 1 – e^-λx, corresponds to the area under the density curve to the left of x.
  - P{X>x} = e^-λx, corresponds to the area under the density curve to the right of x.
  - P{x1<X≤ x2} = e^-λx1 – e^-λx2, corresponds to the area under the density curve between x1 and x2.

Refer: ENGINEERING STATISTICS HANDBOOK (NIST)