Elementary Statistics and Computer Application

Theoretical Distributions

Theoretical distributions are

Discrete Probability distribution
Bernoulli distribution

• A random variable x takes two values 0 and 1, with probabilities q and p ie., p(x=1) = p and p(x=0)=q, q-1-p is called a Bernoulli variate and is said to be Bernoulli distribution where p and q are probability of success and failure. It was given by Swiss mathematician James Bernoulli (1654-1705)

Example

• Tossing a coin(head or tail)
• Germination of seed(germinate or not)
Binomial distribution

• Binomial distribution was discovered by James Bernoulli(1654-1705). Let a random experiment be performed repeatedly and the occurrence of an event in a trial be called as success and its non-occurrence is failure. Consider a set of n independent trails( n being finite), in which the probability p of success in any trail is constant for each trial. Then q=1-p is the probability of failure in any trail.
• The probability of x success and consequently n-x failures in n independent trails. But x successes in n trails can occur in ncx ways. Probability for each of these ways is pxqn-x.
P(sss…ff…fsf…f)=p(s)p(s)….p(f)p(f)….
=p,p…q,q…
=(p,p…p)(q,q…q)
(x times) (n-x times)

Hence the probability of x success in n trials is given by

nc_x p^xq^n-x


Definition

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

The two independent constants n and p in the distribution are known as the parameters of the distribution.