Discussion Problem_(16)

[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) Let f: X -> Y be a function, and let x, x' be elements of X such that f(x) = f(x'). What do we need about f to conclude that x is equal to x'?
(i) Nothing; this is true for all functions f.
(ii) We need f to be one-to-one.
(iii) We need f to be invertible.
(iv) We need f to be onto.
(v) We need f(x) and f(x') to lie in Y.
(vi) We need f to be continuous.
(vii) We need f to be always increasing or always decreasing.

[MI]

Q2

F must be one-to-one

If F is continuous then it must be either increasing or decreasing. If F is discrete no such condition is required.

[Sumit]

Q1. I think it's d) None of the above coz we cant say about it until we aware of value of t n E(x)....But I'm noe sure about it plz confirm...

Q2. F has to be one to one.

[Sinistral]

Q.1) iii
Q.2) ii & iii

[Sumit]

Hey sinstral, could you show ur work for q1...

[anonymouse]

Hi Sumit,

I just found out that this result is called Markov's Inequality. There is a proof given in Wolfram. I also found this page useful in understanding the proof intuitively.

[duck]

Correct answers:
1) (iii)
2) (ii)
Please find attached the working of (Q1)
Q1.png

[Sumit]

@thanks anonymouse
Thanks duck.....

[maahi]

i do not undesrstand d last 2 steps . can u xplain it to me ?

[duck]

Interpret last two steps as:
If x and y are two positive numbers then, x+y ≥ x