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)`.

Mean deviation for n observation x_1,x_2,……….,x_n from their mean barx is given by …….

prove that if the variance of the observations x_1,x_2,x_3,…….,x_n is sigma^2 then the variance of x_1+a,x_2 +a, x_3 + a,………….., x_n + a is sigma^2 .

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_(3),…,x_(n) are the roots of the equation x^(n)+ax+b=0 , the value of (x_(1)-x_(2))(x_(1)-x_(3))(x_(1)-x_(4))…….(x_(1)-x_(n)) is

If x_(1) and x_(2) are the arithmetic and harmonic means of the roots fo the equation ax^(2)+bx+c=0 , the quadratic equation whose roots are x_(1) and x_(2) is

Let X be a random variable taking values x_(1), x_(2), x_(3), ......., x_(n) with probabilities p_(1), p_(2),p_(3),………, p_(n) respectively. Then var (X) = ………..

Prove that the curves x^(2)+y^(2)=ax and x^(2)+y^(2)=by are cuts orthogonally.

If 4^(th) term of (ax+1/x)^(n) is 5/2 then find a and n .

The mean and standard deviation of a set of n_1 observation are barx_1 and S_1 , respectively while the mean standard deviation of another set of n_2 observations are barx_2 and S_2 , respectively . Show that the standard deviation of the combined set of (n_1+n_2) observations is given by standard deviation sqrt ((n_1(S_1)^2 + n_1(S_2)^2)/(n_1+n_2) + (n_1+n_2 (barx_1 - barx_2)^2)/(n_1-n_2)^2)