Q1) Let X be a random variable with density function f(x) such that

P(X≥0)= 1.Then for a given number t > 0 which of the following is true

(i) P(X≤t) ≤ E(X)/t

(ii) P(X≥t) ≥ E(X)/t

(iii) P(X≥t) ≤ E(X)/t

(iv) None of the above

Q2) 1. Consider two random variables X and Y, each taking values in {1, 2, 3}. Let their joint PMF be such that for any 1 ≤ x, y ≤ 3,

Pr (X = x, Y = y) = f(x, y) = 0, if (x, y) ∈ {(1, 3), (2, 1), (3, 2)}

Pr (X = x, Y = y) = f(x, y) > 0, otherwise

Then,

(i) X and Y can be independent or dependent depending upon the strictly positive values.

(ii) X and Y are always independent.

(iii) X and Y can never be independent.

Q3) The letters of the word "MOTHER" are permuted, and all the permutations so formed are arranged in alphabetical order as in a dictionary. Then the number of permutations which come before the word "MOTHER" is

(i) 503

(ii) (6!/2) - 1

(iii) 308

:)