Discussion Problem_ (1)

[duck]

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

[VR]

 no. of words with E - 5! - 120
                   with H - 5! - 120
                   with ME- 4!   24
                   with MO - 4!   24
                   with MOE- 3!     6
                    with MOH -3!    6
                    with MOR -3!    6
                    with MOTH- 2!   2 = 308 = ANS
                     
                      with MOTHER = 309

[Mauli]

i am getting option 3 for all the the three questons.
please tell me if i am wrong.

[Mauli]

hello duck:)
i did the second question by taking examples and i could easily eliminate the options and get iii as the ans, and the third was straight fwd.
in the first however, i did it by putting values . it was a very lengthy way and i am not sure wthr i am write or not. please along with the answer key if u could tell how to approach the first ques in a more formal manner.
thankyou

[duck]

Hi.. :)

Well done.
Q1) Hint:
Just use the definition of Expectation. And partition (0,∞) into intervals (0,t) and (t,∞).

[maahi]

wat is d ans to 1 ? is it 4 ?

[duck]

1) (iii)
2) (iii)
3) (iii)