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 any real numbers and n is anypositive integer, then

If the mean of the set of numbers x_1,x_2, x_3, ..., x_n is barx, then the mean of the numbers x_i+2i, 1 lt= i lt= n is

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) .

If each observation of a new data where variance is sigma^(2) increased by lambda , then the variance of the new data is

For (2n+1) observations x_(1), x_(2),-x_(2),..,x_(n),-x_(n) and 0, where all x's are distinct, let SD and MD denote the standard deviation and median, respectively. Then which of the following is always true ?

The mean and variance of n observations x_(1),x_(2),x_(3),...x_(n) are 5 and 0 respectively. If sum_(i=1)^(n)x_(i)^(2)=400 , then the value of n is equal to

A path of length n is a sequence of points (x_(1),y_(1)) , (x_(2),y_(2)) ,…., (x_(n),y_(n)) with integer coordinates such that for all i between 1 and n-1 both inclusive, either x_(i+1)=x_(i)+1 and y_(i+1)=y_(i) (in which case we say the i^(th) step is rightward) or x_(i+1)=x_(i) and y_(i+1)=y_(i)+1 ( in which case we say that the i^(th) step is upward ). This path is said to start at (x_(1),y_(1)) and end at (x_(n),y_(n)) . Let P(a,b) , for a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a,b) . Number of ordered pairs (i,j) where i ne j for which P(i,100-i)=P(i,100-j) is

Show that the function f defined by f(x)= |1-x+|x|| , where x is any real number, is a continuous function.