Let `x_1, x_2, x_3, ,` be `n` values of a variable X. If these values are changed to `x_1+a ,x+2+a , , x_n+a ,w h e r ea in R ,` show that the variance remains unchanged.

Let x_(1),x_(2),x_(3),..., be n values of a variable X . If these values are changed to x_(1)+a,x+2+a,...,x_(n)+a, where a in R show that the variance remains unchanged.

Let x_(1),x_(2),x_(3),..., be n values of a variable X . If these values are changed to x_(1)+a,x+2+a,...,x_(n)+a, where a in R show that the variance remains unchanged.

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.

If x_(1), x_(2), ………,x_(n) are n values of a variable x such that sum(x_(i)-3) = 170 and sum(x_(i)-6) = 50. Find the value of n and the mean of n values.

Let x_(1),x_(2),x_(n) values of variable X and let 'a' be non zero real number.Then prove that the variance of the observations ax_(1),ax_(2),*ax_(n) is a^(2)Var(X). Also,find their standard deviation.

Let X and Y be two variables with correlation coefficient e. If the values of X and Y serires and changed such that the cov (X,Y) remains unchanged while the variance of X and Y becomes 4 times their original values, then find the new correlation coefficient between X and Y.

A r.v. X assumes values 1, 2, 3, …, n with equal probabilities. If Var (X) = E (X), then n =

If X is discrete random variable takes the values x_1, x_2, x_3 , ... x_n then sum_(i=1) P(x_i) = ........

in If the coefficient of x^(r) and x^(r+1) are equal in the expansion of (1+x)^(2n+1), then find the value of r.