Re: ISI ME I, 2011 doubts

Posted by Sinistral on May 09, 2013; 2:00pm
URL: http://discussion-forum.276.s1.nabble.com/ISI-ME-I-2011-doubts-tp7580534p7580538.html

for ques 28 u wont get mean as 1. integrate once again. ans is (c) only

29)
f(x) = k*  nCx,    x=0,1,2,3 ...n
  summing the above from 0 to n and equating it to 1 will give k= 1/2^n

E(X)= Σk*x*nCx
       = k(0+ nC1 + 2*(nC2) + 3(nC3) ... n(nCn) )    ---- (1)
please note (1+x)^n =  nC0 + nC1 x + nC2  x^2 ---  nCn x^n)
                 differentiating both sides:
                 n(1+x)^(n-1) = 0+ nC1  + nC2 (2x) + nC3  (3x^2) --- nCn  (nx^n-1)
                 puttong x=1
                 n2^n-1  =   0+ nC1 +      2*(nC2) +   3(nC3) ...         n(nCn)
                putting the above value in equation (1) gives k(n*2^(n-1))
                 k= 1/2^n, so mean = n/2

make similar adjustments to calculate E(X^2)