If  is the mean of  prove that

The variance of a random varibale x is defined by the expected value of (x-barx)^(2) where bar x is the mean of x prove that variance (x) = E(x^(2))-{E(x)}^(2)

Velocities of a particle executing SHM are u and v at distances x_(1)andx_(2) respectively from the mean position. Prove that the amplitude of the motion is ((v^(2)x_(1)^(2)-u^(2)x_(2)^(2))/(v^(2)-u^(2)))^(1//2)

10 households in a neighbourhood are sorted acccording to their monthly income are given below 16500, 21700, 18600, 21050, 19500, 17000, 21000, 18000, 22000, 75000:- How many families have monthly income less than the mean income? Prove that in such situation this average is suitable or not?

Let P(n) be the statement : “the arithmetic mean of n and (n + 2) is the same as their geometric mean”’. Prove that P(1) is not true. Also prove that if P(n) is true, then P(n + 1) is also true. How does this contradict the principle of Induction ?

If x,y,z are in A.P.and A1 is the arithmetic mean of x,y and A2 is the arithmetic mean of y,z then prove that the arithmetic mean of A1 and A2 isy

If n arithmetic means xare inserted between 1 and 31 such that the 7^(th) mean: the (n+1)^(th) mean =5:9 then prove that n=14

State Coulomb's law in vector form and prove that vecF_21 = - vecF_12 where letters have their usual meanings.

Using Lagrange's mean value theorem prove that, log (1+x) lt x ("when " x gt 0)

Using Lagrange's mean value theorem prove that, e^(x) gt 1 +x