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# Density function and Equation for Binomial Distribution Function

## Define density function. Derive an equation for binomial distribution function

Probability and density function:

Probability density function is the derivation of probability distribution function of random variable x

`fx(X) = d/dx Fx(x)`

It is also called as frequency function. The probability density function of a discrete random variable is often known as probability mass function. Binomial Distribution function:

If the experiment is repeated n times independently with two possible outcomes namely the occurrence of event (failure) then they are called as Bernoulli’s trials.

In every trial the possible outcomes are only two success (p) and failure (q)

P+q=1

No. of trails may be 0,1,2,3…..n and is a random variable X

The Probability of X successes and (n-x) failures in n independent trials i.e; P(X=x)

By multiplication theorem of probability the probability of x successes and (n-x) failures in n trials is        (p x p x ….. x times} x {q x q x …(n-x)times}  i.e, px  qn-x

But this takes place in nCx ways.

Hence P(X=x)= nCx px qn-x  :x=0,1..n

Where  n= no. of trials

p=probability of success

q=probability of failure

x=no. of occurrences of particular event

The Binomial distribution of a random variable x is
Fx(x) = P(X=x) = sigma (k=0..n) ( n  x )px  qn-x

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