Bhaskar

if X and Y are independent normal random variables with parameters μ and σ², prove that X+Y is independent of X-Y.

Let a_1 < a_2 < ... < a_n denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion of a_i and a_j in the permutation if i < j and a_j precedes a_i. For example the permutation 4, 2, 1, 5, 3 has five inversions - (4, 2), (4, 1), (4, 3), (2, 1), (5, 3). Consider now a random permutation of a_1, a_2, .... , a_n (in the sense that all n! permutations are equally likely) and let N denote the number of inversions in this permutation. Also, let

N_i = number of k: k < i and a_i precedes a_k in the permutation

and note that N = ∑ N_i (from i=1 to n)

(i) Show that N_1, N_2, ...., N_n are independent random variables

(ii) What is the distribution of N_i

(iii) Compute E(N) and Var(N).

I've been thinking of this problem for a couple of days. I managed to come up with a distribution in the form of a sum, but I've been unable to simplify it further. number (iii) is quite beyond me. but is there any simple form for the distribution? can anyone help?

if (1 + x + x^2)^n = a_0 + (a_1)x + (a_2)x^2 + ...... + (a_2n)x^2n,

prove that

a_0 + a_3 + a_6 + ... = a_1 + a_4 + a_7 + ... = a_2 + a_5 + a_8 + ....

and what was the prank and how did you feel after it?

test the series

sum [1 / (ln(n))^(ln(n))] from n=2 to infinity

for convergence or divergence.

just curious about whether

lim(n^n/n!) as n-->∞

exists.

if so, what is it?

I've decided to post these two problems together so as to save points.

here they are:

1) a billiard ball of radius r is struck by a cue at a height h above the centerline. the ball takes off with a velocity of v0, but, due to its 'forward english' finally acquires a speed of (9/7)v0. show that

h = (4/5) R.

2) a simple pendulum of length l and mass m is suspended in a car that is moving with velocity v in a circle of radius R executes small oscillations about its mean position. find the frequency of oscillations.

if a, b, c, .......... are p positive integers whose sum is n, prove that the least value of

a! b! c!........... is q!^(p-r) * (q+1)!^r

a billiard ball initially at rest is given a sharp impulse by a cue. the cue is held a distance h above the centerline. the ball leaves the cue with a speed v0 and, because of its 'forward english', eventually acquires a final speed of (9/7)v0. show that:

h = 4/5 r,

where r is the radius of the ball.

a spring is compressed and its ends tied together with a string. it is then placed in an acid and dissolves.what happened to the stored potential energy in the spring?

if c0, c1, c2,.... cn are the coefficients in the expansion of (1+x)^n, prove that:

c1 - (c2)/2 + c3/3 - c4/4 + ........+ (-1)^(n-1) cn/n = 1 + 1/2 + 1/3 + ...... + 1/n

if a0, a1, a2, ... are the coefficients of x^0, x^1, x^2, .. in the expansion of

(1 + x + x^2)^n, prove that

(a0)^2 - (a1)^2 + (a2)^2 - ... + (-1)^(n-1)*(a_n-1)^2 = an/2 * {1 - (-1)^n*an}

the circle inscribed in triangle ABC touches the sides BC, CA and AB in the points A1, B1, C1; similarly, the circle inscribed in A1B1C1 touches the sides in A2, B2 and C2, and so on. if An, Bn, Cn be the nth triangle so formed, prove that its angles are :

π/3 + (-2)^-n (A - π/3) ,

π/3 + (-2)^-n (B - π/3) ,

π/3 + (-2)^-n (C - π/3)

hence prove that the triangle so formed is ultimately equilateral.

a massless rope is strung over a frictionless pulley. a monkey holds on to one end of the rope and a mirror, having the same weight as the monkey, is attached to the other end of the rope at the monkey's level. can the monkey get away from his image seen in the mirror :

a) by climbing up the rope, b) by climbing down the rope, c) by releasing the rope.

if c1, c2, ..., cn denote the coefficients in the expansion of (1+x)^n, where n is a positive integer, prove that:

c1 – (c2)/2 + (c3)/3 – (c4)/4 + ..... + (-1)^(n-1)(cn)/n = 1 + 1/2 + 1/3 + ... + 1/n

a spring is compressed and its ends tied tightly together. it is then placed in an acid and dissolves. what happens to the potential energy stored in the spring?

let (7 + 4*sqrt(3))^n = p + b,

where n and p are integers and b is a proper fraction. prove that

(1 - b)(p + b) = 1.

let Sn denote the sum of the first n natural numbers. prove that

2 (s1*s2n + s2*s(2n-1) + s3*s(2n-2) + ... + sn*s(n+1) ) = (2n+5)! / 4! (2n-1)!.

a, b and c are real numbers such that

a^2 + b^2 + c^2 = 1.

prove that

-1/2 <= ab + bc + ac <= 1.