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.

Mean deviation for n observations x_(1), x_(2),….., x_(n) from their mean bar(x) is given by :

The average of n numbers x_(1), x_(2), ….., x_(n) is M. If x_(n) is replaced by x', then new average is :

If bar(X) is arithmetic mean of x_(1), x_(2), … x_(n) and a ne bar(X) is any number, then :

Prove that the function f(x)= x^(n) is continuous at x= n, where n is a positive integer.

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) .

Let x_(1), x_(2),….., x_(n) be n observations and x be their arithmetic mean. The formula for the standard deviation is given by :

If A = (x: x = (1)/(y), y in N ), where N is the set of natural numbers , then

Show that the middle term in the expansion of (1+x)^(2n) is 1.3.5…(2n-1)/n! 2n.x^n , where n is a positive integer.

Sum of non - negative integral solutions of : x_1+x_2+.......+x_nle x (where n is +ve integer ) is :