Binomial Expansion
( x + y )^{n} =^{ }^{ r=n}Σr=0 ( ^{n}Cr ) x^{(n-r) } y^{r}
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) = ( ^{n}Cr ) (1-y)^{(n-r) } y^{r}
Where b(r ; n,x) represents Binomial Distribution function whose mean μ =ny ; and variance σ^{2}=ny(1-y)
Total number of terms | n +1 | |
Numerical value of expression | ( x + y )^{n} | |
Co-efficient of rth term of y | ( ^{n}Cr ) | |
Value of x | x^{(n-r)} | |
Value of y | y^{r} | |
Product of x & y | x^{(n-r) } y^{r} | |
Total value of rth term of y | ( ^{n}Cr ) x^{(n-r) } y^{r} | |
Sum of all Co-efficients in expansion | ^{ r=n}Σr=0 ( ^{n}Cr )=2^{n} | |
Sum of square of all Co-efficients in expansion | ^{ r=n}Σr=0 ( ^{n}Cr )^{2} = ^{ 2n}Cn | |
Mean for probability distribution (if that is the case) | μ =ny | |
Variance for probability distribution (if that is the case) | σ^{2}=ny(1-y) |