Binomial Expansion

The expansion is of the type (x + y )n  where x, y are two real numbers & n is an integer.

We have (x + y )=  nC0xn y0 + nC1 xn-1y1 + nC2 xn-2y2 + nC3 xn-3y3  +.nC4 xn-4y+...........+ nCn-1 x1yn-1 + nCn x0yn

 Or         (x + y )= Σ k=0 --> n nCk xn-k y where nCis also written as . These are called binomial coefficients and are positive numbers.

1.  There are n+1 terms in the expansion. If n is even , no. of terms is odd & vice versa. In case of even terms, all coefficients occur in pairs. In case of odd terms , only the middle coefficient does not occur in pair.

2. Arranging co-efficients in rows for successive values of k from k=0 to k =n gives rise to Pascal's triangle .

3. nCk = n! / [(k!)*(n-k)!] . Therefore nC0   = 1 & nC= 1 & 0Ck  = 0 for all k > 0 .

4. nCk = n-1Ck -1   + n-1Ck   for n , k >0

5. nCk = i=1 --> k   [n - (k-i)]/i

 

Pascal's Triangle


The expansion ( x + y ) n =xn + nC1xn-1y + nC2xn-2y2 + nC3xn-3y3 +...+ nCrxn-ryr + ...... +nCn-1x1yn-1 +nCnyn ;

The Coefficients form what is known as Pascal's Triangle. The characteristics are:-
1) No. of total terms is n+1. It starts with n=0 and ends with n=n.

2) Coefficients of n=0 and n=1 are 1.

3) The coefficients of any row can be obtained by adding the 2 numbers in the preceding row that are just at the left and just at the right of the given number.

4) Coefficients that are equidistant from the beginning and the end are equal

5) nCr = nCn-r

6) to view the triangle click below

7) workout on factorials 4

8) binomial co-efficient chart

9) binomial calculation click


Combinations:

The general formula is:

n C k = n! / [k! (n-k) !].

Combination of n objects in a group of size k, k n
n
k
To find value of k th term in expansion of ( x + y )n
put the value of n,k above, press calculate and then press CLICK below
n value
              k value                                     
     Value of k th term is  x             y



Combination of n objects in a group of size k, repetitions allowed ( n+k-1Ck )
n
k
 

Probability Mass Function

Probability distribution of a random discrete variable is called probability mass function whereas probability distribution of a random continuous variable is called a probability density function.

If k is a random discrete variable and there are n number of trials, and there are only 2 possible outcomes- success (probability-p) & failure (probability-q=1-p), then probability of getting k no. of successes in n number of trials is as per Binomial Distribution --

P(p)=         nCpk  qn-k

B(k)=         nCpk  qn-k   where B(k) is called Binomial frequency function of p. ΣB(k)= (p+q)n  for all values of k =0 to n. In other words, the

                sum of the values of frequency at point k=0,1,2,.....n is equal to (p+q)n

B(k) / B(k-1) =((n+1-k)/k)*p/q =Q

Qk > 1 when  (n+1)p > k(p+q)  i.e frequency function value at that k is greater than its previous k value.

Qk = 1 when  (n+1)p = k(p+q) i.e frequency function value at that k is equal to its previous k value.

Qk < 1 when  (n+1)p < k(p+q) i.e frequency function value at that k is less than its previous k value.

Most Probable Frequency : Supposing k has values   n( n0,n1,n2,.......nn , )then if for any ni there is a nj  such that B( nj) B( ni) where i,j =0,1,2,3.....n, then nj is called the most probable value of k, corresponding frequency B(nj) being most probable frequency.

Uniform Sample Space: If we toss 2 coins, the sample points are (head ,head),(head,tail),(tail,head),(tail, tail). Each of the 4 sample points have the same probability i.e 1/4. Hence sample space is uniform. A sample space is said to be uniform if each point has the same probability.

Non-Uniform Sample Space: A sample space is said to be non-uniform if each point in it does not have the same probability. For example, no. of heads in tossing of 4 coins. These are nC0 , nC1 ,nC2 , nC3, nC4,  which represent probabilities for no head, 1 head, 2 head, 3 head, 4 head respectively and  have different values. Hence, the sample space is non-uniform.

For large value of independent trials n: When number of independent trials n -->infinity, binomial distribution function tends to be the normal distribution function.

Stirling's approximation: click