Likelihood distribution and Approximations

WEEK 8

TOPIC:- Likelihood distribution and Approximations

Content material: (i) Binomial approximations by Poisson distributions

(ii) Poisson distribution

1. (iii) Regular distributions and Regular approximations
2. (iv) Binomial approximations by Poisson distributions

Binomial Distributions

In our final dialogue in sss2, we leant about idea of chance and calculated chance and occasions.

We wish to see find out how to discover possibilities of various values of discrete variables comparable to counts scores.

Likelihood Distributions :- Sure pure occurrences have attribute random behaviours. When theoretical chance mannequin or distribution is constructed on their nature of randomness, we are saying such a mannequin or theoretical distribution is known as a chance distribution.

Mathematically that is outlined :- when a random variable x can assume a discrete set of values x₁,x₂,………x_okay with chance p₁,p₂,p₃…….p_okay the place then we are saying {that a} chance perform of x written as p(x), is outlined.

Keep in mind the binomial expression, the binomial chance lings around the binomial theorem. Think about repeated and impartial trials of an experiment with two doable outcomes, one of many end result is known as success and the opposite failure. Let P be the chance of success then

q=1-p is chance of failure. The curiosity is in variety of successes not the order they happen when n numbers of trials are made and x is numbers of successes, then the binomial distribution is given by

The place x=0,1,2………..n

And when n and p are fixed then the perform B(x;n,p) is discrete chance distribution with values as comply with within the desk.

The successive values of p(x=x) might be seen to be the identical as these to the binomial growth of . Thus

Instance 1:- write out the values of a binomial distribution for a toss of 8 cash at a time and let x be the variety of heads that seem. Draw the histogram, corresponding frequency polygon of this distribution after which decide the chance distribution of x.

Resolution :-

of trials, A=8, Let head be successful with chance,

B