prove that if the variance of the observations `x_1,x_2,x_3,…….,x_n` is `sigma^2` then the variance of `x_1+a,x_2 +a, x_3 + a,………….., x_n + a` is `sigma^2` .

Mean deviation for n observation x_1,x_2,……….,x_n from their mean barx is given by …….

Given that bar(x) is the mean and sigma^(2) is the variance of n observation x_(1), x_(2), …x_(n). Prove that the mean and sigma^(2) is the variance of n observations ax_(1),ax_(2), ax_(3),….ax_(n) are abar(x) and a^(2)sigma^(2) , respectively, (ane0) .

S.D. of the observation x_1, x_2, x_3 ..... x_n is 3.5 then S.D. of -2x _1-3,-2x_2-3,.........-2x_(n)-3 is ….........

variance of 6 observation is 250 then sum(x_i-barx)^2=1200.

If each of the observation x_(1), x_(2), ...,x_(n) is increased by 'a', where a is a negative or positive number, show that the variance remains unchanged.

x_1,x_2,x_3,….x_n then (x_1-barx)+(x_2-barx)+(x_n-barx)=0.

Prove that the coefficient of x^n in the expansion of (1+x)^(2n) is twice the coefficient of x^n in the expansion of (1+x)^(2n-1) .

Let n and k be positive such that n leq (k(k+1))/2 .The number of solutions (x_1, x_2,.....x_k), x_1 leq 1, x_2 leq 2, ........,x_k leq k , all integers, satisfying x_1 +x_2+.....+x_k = n , is .......

Let X be a random variable taking values x_(1), x_(2), x_(3), ......., x_(n) with probabilities p_(1), p_(2),p_(3),………, p_(n) respectively. Then var (X) = ………..