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)
| 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) |