Download Probability Theory

Probability theory
End Semester Examination
KV
(1) Show that the distribution function of a random variable X is right
continuous.
4 marks
(2) Let An , n ≥ 1 be an infinite sequence of independent events from a
probability space. We are interested in finding out how many of the
An occur.
(a) Show that B(∪N
i=n Ai ) = 1 −
Qi=N
i=n
(1 − P(Ai )).
∞
(b) Show that ∩∞
i=1 [∪j=i Aj ] is the event of infinitely many An ’s occuring.
P
(c) Suppose n P(An ) = ∞. Show that the probability of the even
∞
−x
∩∞
i=1 [∪j=i Aj ] is 1. (use 1 − x ≤ e )
5 marks
(3) In this problem we outline the proof of the strong law, under the assumption on the fourth moments. Follow the steps and complete the
proof.
Let Xi be a sequence of independent identically
distributed random
Pn
4
variables. Suppose E(Xi ) < ∞. Let Sn = i=1 Xi .
(a) State the strong law of large numbers.
(b) Argue that we may assume E(Xi ) = 0
(c) Show that P( Snn | ≥ ) ≤
4)
E(Sn
4 n4
(d) Examine the terms in the expression of E(Sn4 ) and argue that the
only terms which survive are terms which are fourth powers and
products of squares.
1
(e) Show that E(Xi2 ) ≤ (E(Xi4 )) 2 .
1
8 marks
(4) We say a sequence {Sn : n ≥ 1} is a martingale with respect to a sequence of random variables {Xn : n ≥ 1} if for all n, E(Sn+1 |X1 , . . . Xn ) =
Sn . Let Xn be a sequence of independent random
variables with P(Xi =
Pn
α
Xi
i=1
. For what values
1) = p and P(Xi = −1) = 1 − p. Let Sn = ( p )
of α is the sequence Sn a martingale ?
6 marks
(5) Let X be a random variable such that P(X ∈ Z) = 1. Show that
the characteristic definition of X is a periodic function with period
2π. Show that if the characteristic function satisfies φX (2π) = 1 then
P(X ∈ Z) = 1
6marks
(6) Four fair die are thrown and we define X to be the sum of the four
numbers which show up.
(a) Write the generating function of the random variable X
(b) Compute the probability that X is 10, using the above generating
function.
5 marks
(7) Find the density of the random variable with the characteristic function e−|t|
5
marks
(8) A perfect die is thrown 200 times and the sum of the numbers obtained is calculated. What is the probability that the value obtained
is between 670 and 740? Use the central limit theorem and table of
N (0, 1) provided.
5 marks
(9) Let X and Y be independent N (0, 1) random variables and let Z =
X + Y . Find the distribution of Z, given X > 0 and Y > 0.
6 marks
2