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 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) .

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

The mean of n observations is bar(x) . If the first observation is increased by 1, the second by 2 and so on, the new mean is . . . .

Draw the x to t graphs for positive, negative and zero acceleration.

If A_(1),A_(2),A_(3),…,A_(n) are n points in a plane whose coordinates are (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),…,(x_(n),y_(n)) respectively. A_(1)A_(2) is bisected in the point G_(1) : G_(1)A_(3) is divided at G_(2) in the ratio 1 : 2, G_(2)A_(4) at G_(3) in the1 : 3 and so on untill all the points are exhausted. Show that the coordinates of the final point so obtained are (x_(1)+x_(2)+.....+ x_(n))/(n) and (y_(1)+y_(2)+.....+ y_(n))/(n)

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.

Statement-1: If N the number of positive integral solutions of x_(1)x_(2)x_(3)x_(4)=770 , then N is divisible by 4 distinct prime numbers. Statement-2: Prime numbers are 2,3,5,7,11,13, . .

Show that the function given by f(x)=e^(2x) is increasing on R.