There are basically two types of random variables, called continuous and discrete random variables. We shall discuss the probability distribution of the discrete random variable. The discrete random variable may be define as the random variable that is countable in nature. Like, the number of heads, number of books, etc.
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The probability distribution that deals with this type of random variable is called the probability mass function (pmf).
There are various types of discrete probability distribution. They are as follows:
A random variable X is said to have a discrete probability distribution called the discrete uniform distribution if and only if its probability mass function (pmf) is given by the following:
P(X=x)= 1/n , for x=1,2,3,….,n
0, otherwise.
A random variable X is said to have a discrete probability distribution called the Bernoulli distribution if and only if its probability mass function (pmf) is given by the following:
P(X=x)=px (1-p)1-x, for x=0,1.
0, otherwise.
A random variable X is said to have a discrete probability distribution called the Binomial distribution if and only if its probability mass function (pmf) is given by the following:
P(X=x)=nCx pxqn-x, for x=0,1,2,….n; q=1-p.
0, otherwise.
A random variable X has a discrete probability distribution called the Binomial distribution if and only if its probability mass function (pmf) present the following:
P(X=x)=nCx pxqn-x, for x=0,1,2,….n; q=1-p.
0, otherwise.
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A random variable X has a discrete probability distribution called Poisson distribution if and only if its probability mass function (pmf) present by the following:
P(X=x)=e-k kx , for x=0,1,2,….; k>0
0, otherwise.
A random variable X has a discrete probability distribution refers to the negative binomial distribution if and only if its probability mass function (pmf) present by the following:
P(X=x)=x+r-1Cr-1 pr qx , for x=0,1,2,….
0, otherwise.
A random variable X has a discrete probability distribution refers to the geometric distribution if and only if it is the following:
P(X=x)=qx p , for x=0,1,2,….; 0<p<= q=”1-p.
0, otherwise.
A random variable X has a discrete probability distribution refers to the the hyper geometric distribution with the parameters N, M and nif it assumes only non negative values with the probability mass function as the following:
P(X=k)=MCk N-MCn-k / NCn, for k=0,1,2,….min(n,M).
0, otherwise.
Here, N is a positive integer. M is also a positive integer that does not exceed N and the positive integer n at most of N.
The generalization of the discrete probability distribution called the binomial distribution. The generalized binomial distribution refers to the multinomial distribution and present in the following manner:
If x1,x2,…. xk are k types of random variables, then they has the discrete probability distribution as the following:
p(x1,x2,…. xk)= (n!/ x1!x2!…. xk!) p1x1 p2x2….. pnxn, for k=0,1,2,….min(n,M).
A discrete random variable X follow a discrete probability distribution called a generalized power series distribution if its probability mass function (pmf) present by the following:
P(X=x)= ax hx/f(h); x=0,1,2…. ; ax>=0
0, elsewhere.
We must note that in this discrete probability distribution, f(h) is a generating function s.t:
f(h)= ax hx , h>=0
so that f(h) is positive, finite and differentiable and S is a non empty countable sub-set of non negative integers.