Binomial Expansion

( x + y )n =  r=nΣr=0 ( nCr ) x(n-r) yr

Binomial  Distribution of Probability

The beauty of binomial expansion is that if x+y=1, the result of expansion is also 1. Suppose in a chancy event, there are 2 outcomes the probability of outcome one is x & that of outcome two is y, then the probability distribution of  outcomes of the events can be represented by  the various terms of binomial expansion of (x+y) to the power n where n has positive integer value. We call these Binomial distribution of probability. Here x=1-y. Taking r as a variable which indicates  rth term of expansion & r=0,1,2,3.....n and taking 0< y < 1, we can write b(r ; n,x) = ( nCr ) (1-y)(n-r) yr

Where  b(r ; n,x) represents Binomial Distribution function  whose mean  μ =ny ; and variance σ2=ny(1-y)

 Value of     x Value of     y Value of     n Value of     r

 Total number of terms n +1 Numerical value of expression ( x + y )n Co-efficient of rth term of y ( nCr ) Value of x x(n-r) Value of y yr Product of x & y x(n-r) yr Total value of rth term of y ( nCr ) x(n-r) yr Sum of all Co-efficients in expansion r=nΣr=0 ( nCr )=2n Sum of square of all Co-efficients in expansion r=nΣr=0 ( nCr )2 = 2nCn Mean for probability distribution (if that is the case) μ =ny Variance for probability distribution (if that is the case) σ2=ny(1-y)