Discreet probability distribution is that type of distribution that deals with discreet types of random variables. The discreet random variables deal with different kinds of discreet probability distribution.
A uniform distribution is a discreet probability distribution over the range [1,n] if its probability mass function (pmf) is given by
P(X=x)=1/n, for x = 1 …. n
Here ‘n’ is known as the parameter of the discreet probability distribution. This discreet probability distribution lies in the set of all positive integers. This type of discreet probability distribution is appropriate in the case of die experiment, an experiment with a deck of cards, etc.
A random variable X is said to have a discreet probability distribution called bernoulli distribution with parameter ‘p’ if its probability mass function (pmf) is given by
P(X=x)= px (1-p)1-x, for x = 0,1
In this type of discreet probability distribution, the parameter ‘p’ satisfies two values i.e. 0 and 1. This type of discreet probability distribution is appropriate for random experiments whose outcomes are of two types.
A random variable X is said to have a discreet probability distribution called binomial distribution if it assumes non negative values and its probability mass function (pmf) is given by
P(X=x)=(ncx) pxqn-x, x = 0,1 …. n;q=1-p
The tossing of a coin is a very popular experiment which deals with the theory of this type of discreet probability distribution.
A random variable X is said to have a discreet probability distribution called poisson distribution only if its probability mass function (pmf) is given by
P(X=x)=e-α αx/x!, x=0,1,……
In this type of discreet probability distribution, α is the parameter. This discreet probability distribution can be used in cases like the number of faulty blades in a packet of 100, the number of suicides reported in a particular city, etc. The discreet probability distribution can also be used for obtaining the number of printing mistakes on each page of a book, the number of cars passing a crossing during the busy hours of a day, the number of airplane accidents in some unit of time, the emission of radioactive (alpha) particles, etc.
A random variable X is said to have a discreet probability distribution called geometric distribution only if its probability mass function (pmf) is given by
P(X=x)=qx p; x=0,1, …..
This type of discreet probability distribution is applied in those types of experiments where there is a series of independent trials, and for each the probability of success ‘p’ remains constant.
A random variable X is said to have a discreet probability distribution called hyper geometric distribution only if its probability mass function (pmf) is given by
P(X=x)= Mck N-Mcn-k / Ncn ; k= 0 , 1 , ….. , min(n,M)
In this type of discreet probability distribution, ‘N,’ ‘M’ and ‘n’ are parameters and are positive integers. This type of discreet probability distribution is useful in the following experiment.
Consider an urn with N balls, M of which are white and (N-M) are red. If ‘n’ balls are drawn at random without replacement, then the probability of getting ‘k’ white ball out of ‘n’ balls is that discreet probability distribution.
This type of discreet probability distribution, called multinomial distribution,is the generalization of binomial distribution. So, a random variable X is said to follow this discreet probability distribution only if
p(x1, x2, …. , xk)=(n!/ x1!, x2!, …. , xk!) p1x1,p2x2, ….. , pkxk , ∑xi= n and ∑pi=1, i=1, .. , n
This type of discreet probability distribution is used in those types of experiments where there will be ‘n’ repeated trials and each trial will have a discreet number of possible outcomes.
