Probability Distributions
Probability distribution is the distribution or representation of values of that random variable may have. It represent how likely one can could find the possible values of random variable.
Suppose you draw a random sample and measure the weights of the different people. You can create a distribution of weights. The distribution is useful when you need to know which outcomes are most likely, the spread of values, and the likelihood of different results.
Types
Bernoulli Distribution
The simplest distribution is Bernoulli Distribution. It has only two possible outcome, either 1 (success) or 0 (failure), and a trial.
Let X be the random variable of having fight between A and B that takes 1 as the probability of winning of A as success (p) and 0 as the probability of losing of A as failure (1-p or q).
- Probability of success = 0.4
- Probability of gailure= 0.6
PMF is given :
Distribution :
Uniform Distribution
A uniform distribution is the distribution of probabilities that are same at each point in the interval. It has the shape like rectangle, also called rectangle distribution.
PMF is gives as :
Here,
a is location parameter.
b is scale parameter.
Distribution :
Binomial Distribution
A distribution in which there are only two possible outcomes, the probability of success and probability of failure of an event repeated multiple times and each time the event is independent.
PMF is given as :
Distribution :
Normal Distribution
The distribution which is like ‘bell curve’ having no bias either on left side or right side is known as normal distribution. This distribution is found most commonly in nature.
The Normal Distribution has:
- mean = median = mode
- symmetry about the center
- 50% of values less than the mean and 50% greater than the mean
Example :
-heights of people
-blood pressure
-marks on a test
PDF is given as :
Distribution
Poisson Distribution
The distribution in which one successful event does not influence other successful event and the probability of success of short interval is equal to probability of success of longer interval is called Poisson Distribution.
The probability of success in an interval approaches zero as the interval becomes smaller.
PMF is given as :
Here,
µ is the parameter of this distribution and µ = λ.t
where,
λ = the rate at which event occur.
t = length of interval.
Distribution
Exponential Distribution
It is the distribution which is used to describe the time between events in the Poison point process.
PDF is given as :
Distribution :
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