5. THEORETICAL DISCRETE DISTRIBUTIONS

 

5.1     BERNOULLI DISTRIBUTION:

A random variable X which takes two values 0 and 1, with probabilities q and p respectively, i.e., P(X=1)=p, P(X=0)=q, q=1-p is called a Bernoulli variate and is said to have a Bernoulli distribution.

 

5.1.1 Moments of Bernoulli distribution.

The r^th  moment about origin is

μ_r^’ = E(X ^r) = 0^r  . q+1^r.p=p; r=1, 2,….

The moment generating function of Bernoulli variate is given by:

 

5.2  BINOMIAL DISTRIBUTION:

A random variable X is said to follow binomial distribution if it assumes only non-negative values and its probability mass function is given by

P(X=x) =p(x) =

The two independent constants n and p in the distribution are known as the parameters of the distribution. ‘n’ is also sometimes, known as the degree of the binomial distribution.

Binomial distribution is a discrete distribution as X can take only the integral values, viz., 0,1,2,…..,n. Any variable which follow binomial distribution is known as binomial variate.

We shall use the notation X~ B(n,p) to denote that the random variable X follows binomial distribution with parameters n and p.

The probability p(x) is also sometimes denoted by b(x,n,p).

 

5.3  POISSON DISTRIBUTION:

Poisson distribution was discovered by the French mathematician and physicist Simeon Denis Poisson who published it in 1837. Poisson distribution is a limiting case of the binomial distribution under the following conditions.

          (i) n, the number of trials is indefinitely large, i.e., nΰ ∞.

          (ii) p, the constant probability of success for each trial is indefinitely small, i.e., pΰ 0.

          (iii) np = λ, is finite. Thus p = λ/n, q = 1-λ/n, where λ is a positive real number.

         

A random variable X is said to follow a Poisson distribution if it assumes only non-negative values and its probability mass function is given by

otherwise

Here λ is known as the parameter of the distribution.

We shall use the notation X~ P(λ) to denote that X is a Poisson variate with parameter λ.

 

5.4 NEGATIVE BINOMIAL DISTRIBUTION:

A random variable X is said to follow a negative binomial distribution if its probability mass function is given by

,otherwise

Also

                                                     

       

 

 

 

, otherwise

Which is the (x+1)^th term in the expansion of , a binomial expansion with a negative index. Hence the distribution is known as negative binomial distribution.

 

 

5.5 GEOMETRIC DISTRIBUTION:

Suppose we have a series of independent trials or repetitions and on each repetition or trial the probability of success ‘p’ remains the same. Then the probability that there are x failures preceding the first success is given by q^x p.

A random variable X is said to have a geometric distribution if it assumes only non-negative values and its probability mass function is given by

P(X=x) =

P(X=x) = 0, otherwise

 

5.5.1 Lack of Memory.

The geometric distribution is said to lack memory in a certain sense. Suppose an event E can occur at one of the times t=0,1,2,……and the occurrence(waiting) time X has a geometric distribution.

Thus, P(X=t) = q^t .p;  t = 0,1,2,…..

Suppose we know that the event E has not occurred before k, i.e., X ≥ k. Ley Y = X-k. Thus Y is the amount of additional time needed for E to occur. We can show that

P(Y=t/X ≥ k) = P9X = t) = pq^t

Which implies that the additional time to wait has the same distribution as initial time to wait.

Since the distribution does not depend upon k, it, in a sense, ‘lacks memory’ of how much we shifted the time origin. If ‘B’ were waiting for the even E and is relieved by ‘C’ immediately before time k, then the waiting time distribution of ‘C’ is the same as that of ‘B’.

 

5.6 HYPERGEOMETRIC DISTRIBUTION :                

When the population is finite and the sampling is done without replacement, so that the events are stochastically dependent, although random, we obtain hypergeometric distribution. Consider an urn with N balls, M of which are white and N-M are red. Suppose that we draw a sample of n balls at random ( without replacement) from the urn, then the probability of getting k white balls out of n, (k <n) is

A discrete random variable x is said to follow the hypergeometric distribution if it assumes only non-negative values and its probability mass function is given by

P(X =k) =h(k;N,M,n) =  ;     k =1,2,….,min(n,M).

                                      = 0 otherwise

5.6.1 Mean and Variance of the Hypergeometric Distribution:

 

 

5.7 MULTINOMIAL DISTRIBUTION:

This distribution can be regarded as a generalization of Binomial distribution.

When there are more than two mutually exclusive outcomes of a trial, the observations lead to multinomial distribution. Suppose E_1,E_2,….E_k are k mutually exclusive and exhaustive outcomes of a trial with the respective probabilities p₁,p_2…..,p_k.

The probability that E₁ occurs x₁ times, E₂ occurs x₂ times …and E_k, occurs x_k times in n independent observations, is given by

P(x₁,x₂,….x_k) =

Where and c is the number of permutation of the eventsE_1,E_2, ……..E_k,­

To determine c, we have to find the number of permutations of n objects of which x₁ are of one kind, x₂ of another kind,…. And x_k kind which is given by

                 ,

                                   

 

Which is the required probability function of the multinomial distribution.

Since total probability is 1, we have

5.8 DISCRETE UNIFORM DISTRIBUTION:

A random variable X is said to have uniform distribution on n points {x_1,x_2,……..,x_n} if its p.m.f. is given by:

P(X=x_i) = ;        i=1,2,……….,n

For example, if X has a uniform distribution on the points { 0,1,2,…,n}, then P(X =i)  = ;       i=0,1,2,…..,n        

Such distributions can be concelved in practice if under the given experimental conditions, the different values of the random variable become equally likely. Thus for a die experiment, and for an experiment with a deck of cards such distribution is appropriate.