Let `x_1,x_2,x_3,…x_n` be n values of a variable x. If these values are changed to `x_1+a, x_2+a,…x_n+a`, where a∈ 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.

Let x be a variable assuming values x_1 , x_2 ,……. x_n state the condition when AM(x) = GM(x) = HM(x)

If the values of variable X are x_(1),x_(2),.......x_(n) , then the variance of ax_(1),ax_(2),.......ax_(n) (a is any non-zero real number) is

Let x_1,x_2,……,x_n be n observations, and let barx be their arithimetic mean and sigma^2 be their variance. Statement1: Variance of 2x_1, 2x_2,…..,2x_n is 4 alpha^2 Statement2: Arithmetic mean of 2x_1, 2x_2,…..,2x_n is 4 barx .

Let x_(1), x_(2),….,x_(n) be n observations, and let barx be their arithmatic mean and sigma^(2) be their variance. Statement 1 : Variance of 2x_(1),2x_(2),….,2x_(n) is 4sigma^(2) Statement 2 : Arithmatic mean of 2x_(1), 2x_(2),….,2x_(n) is 4 bar x .

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) .

For what value of k will the function f(x) =kx x=1,2,3 ...., n be a probability distribution of a random variable ?

If in the expansion of (1+ x )^(m) (1-x) ^(n) the coefficient of x and x ^(2) are 3 and (-6) respectively, then the value of n is-

A variable assumes the values x_1 , x_2 ,…….., x^_n with frequencies f_1 , f_2 ,………, f_n In respectively, show that overset(n)underset(i=1)sum f_i(x_i - barx) = 0 .

If the constant term in the binomial expansion of (x^2-1/x)^n ,n in N is 15, then find the value of n .