Quantitative Results

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 is defined 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)=p^{x} (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)=^{n}C_{x }p^{x}q^{n-x}, for x=0,1,2,….n; q=1-p.

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)=^{n}C_{x }p^{x}q^{n-x}, for x=0,1,2,….n; q=1-p.

0, otherwise.

A random variable X is said to have a discrete probability distribution called Poisson distribution if and only if its probability mass function (pmf) is given by the following:

P(X=x)=e^{-k} k^{x} , for x=0,1,2,….; k>0

0, otherwise.

A random variable X is said to have a discrete probability distribution called the negative binomial distribution if and only if its probability mass function (pmf) is given by the following:

P(X=x)=^{x+r-1}C_{r-1} p^{r} q^{x} , for x=0,1,2,….

0, otherwise.

A random variable X is said to have a discrete probability distribution called the geometric distribution if and only if it is the following:

P(X=x)=q^{x} p , for x=0,1,2,….; 0<p<= q=”1-p.

0, otherwise.

A random variable X is said to have a discrete probability distribution called 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)=^{M}C_{k} ^{N-M}C_{n-k }/ ^{N}C_{n}, 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.

There is also the generalization of the discrete probability distribution called the binomial distribution. That generalized binomial distribution is called the multinomial distribution and is given in the following manner:

If x_{1},x_{2},…. x_{k} are k types of random variables, then they are said to have the discrete probability distribution as the following:

p(x_{1},x_{2},…. x_{k})= (n!/ x_{1}!x_{2}!…. x_{k}!) p_{1}^{x}_{1} p_{2}^{x}_{2}….. p_{n}^{x}_{n}, for k=0,1,2,….min(n,M).

A discrete random variable X is said to follow a discrete probability distribution called a generalized power series distribution if its probability mass function (pmf) is given by the following:

P(X=x)= a_{x} h^{x}/f(h); x=0,1,2…. ; a_{x}>=0

0, elsewhere.

It should also be noted that in this discrete probability distribution, f(h) is a generating function s.t:

f(h)= a_{x} h^{x} , h>=0

so that f(h) is positive, finite and differentiable and S is a non empty countable sub-set of non negative integers.