some university probability problems

ˆ I expect questions to be solved from rst principles.

ˆ I need you to justify steps, not just write the nal answer. I am more interested in seeing whether you have the correct understanding of the concepts.

ˆ Please do not use the internet (except Wolfram Alpha).

1.An experiment consists of throwing a 4-sided dice with numbers f1;2 ;3 ;4 g repeatedly till the number 3 shows up for the third time. If the number 3 shows up for the third time on the lth throw, assign each such outcome a probability 1 ( l 1 2 ) 3l 3(1 p)p l 3 ; l 3.

(a)(1 point) Let be the set of all possible outcomes of the experiment. Does have a nite cardinality or an in nite cardinality?

(b)(3 points) Let Abe the event that sum of all the numbers in the experiment adds to 15.

Describe the elements of Aand calculate the probability of A.

2.(2 points) Random variables A; B; Ctake values in some nite set S, and we are also given that P(A = a; B =b; C =c) > 0; 8a; b; c: Finally we are given that A! B! Cand A ! C! Bare Markov chains. Show that Ais independent of B; C. IE Dept, CUHK [email protected] Page 1 of 4 Probabilty Theory and Applications: Final Exam Online| Spring 2021, IERG 2470B/ESTR 2308 3.(6 points) Xand Yare jointly Gaussian random variables with mean zero and variance 1 and 2 respectively. We are also given that E( X Y) = 1 2 . Therefore the covariance matrix of X; Yis given by K=" 1 1 2 1 2 2# :

The joint distribution is then given by fX Y ( x; y ) = 1 2 p det( K)exp 1 2 h x y i K 1" x y #!

:

Determine the conditional pdf of Xconditioned on X+Y = 6.

4.There is a collection of n+ m tickets that are numbered f1 ; ::; n +mg, where n > m; n; m 2N, (the numbering is not relevant). A subset of size nof these tickets are colored red and the rest are colored blue. A random experiment consists of picking out all the tickets one by one.

Assume a uniform probability model on these outcomes.

Let Adenote the event that for any time, 1 k n+ m the number of tickets colored red that have been picked out is strictly larger than the number of blue colored ones. We will use the following steps to calculate P( A).

Construct the following collection of random variables:

X i= 1 if the ith ticket drawn is colored red, else set X i= 1. Now, de ne a sequence of -algebras as follows:

F 1= (X n+ m ) ; F 2= (X n+ m ; X n+ m 1) ; :::; F m+n = (X m+n; X m+n 1; ::; X 1).

De ne Yl= n m (X n+ m + +X n+ m l+1 ) n + m l ; 1 l n+ m 1 and Y 0 = n m n + m .

(a)(3 points) Show that Y l is a Martingale.

(b)(2 points) De ne a bounded stopping time T= min ffl:Y l = 0 g; n +m 1g: Use the optional stopping time theorem to show that P(A ) = n m n + m .

(c)(2 points) Let Bdenote the event that the rst three tickets are colored red. Use the above part to determine P( A \ B). (Assume for this part that n 3.) IE Dept, CUHK [email protected] Page 2 of 4 Probabilty Theory and Applications: Final Exam Online| Spring 2021, IERG 2470B/ESTR 2308 5.Let Xbe an Poisson random variable distributed according to the probability mass function, denoted by Poi( ), pX ( X =k) := k k !e ; k 0:

For n 1, de ne (X ) n := E X (X 1)::(X (n 1)) :

(a)(1 point) Compute ( X) n for n 1.

(b)(2 points) Use the above part to compute E( X5 ):

(c)(1 point) Compute the characteristic function E( eitX ).

(d)(2 points) Let X 1; ::; X n; ::

be a countably-in nite sequence of independent random vari- ables (de ned on the same ) where X i Poi 1 i ( i+1) ; i 1. De ne S n = n X i =1 X i:

Compute the distribution of S n.

(e)(2 points) Let A be de ned as A=f! : lim n!1 S n exists and is nite :g Prove that P( A) = 1.

6.(6 points) Three football teams A; B; Cplay a sequence of matches (only two teams play in a match). After every match the losing team (there are no ties) gets replaced by the waiting team (unless the winning team has played three consecutive matches). If a team is playing three consecutive matches, then after the third match, that team is replaced irrespective of the outcome of the match.

You are given that the outcome of each match is independent. When A plays B, A wins with probability 0.6; when A plays C, A wins with probability 0.55; when B plays C, B wins with probability 0.5.

Find the stationary probability that team A is playing in the current game.

(note: You have to come up with the right states and create a Markov chain and then calculate its stationary distribution. Finally using its stationary distribution, you have to arrive at the answer). IE Dept, CUHK [email protected] Page 3 of 4 Probabilty Theory and Applications: Final Exam Online| Spring 2021, IERG 2470B/ESTR 2308 7.(5 points) A random process fX g t, t 0, (and X 0= 0) is said to have independent increments , if for all k 2 and for all 0 < t 1< t 2< :: < t kthe random variables ( X t1 ; X t2 X t1 ; :::; X tk X tk 1) are mutually independent. Further, a random process fX g t, t 0, (and X 0= 0) is said to be stationary , if the distribution of X t2 X t1 depends only on t 2 t 1 .

Consider a random process fX g t, t 0, (with X 0= 0) that takes values in N[ f 0g , and for every !,X t( ! ) is non-decreasing in t. Further assume that the random process has independent increments and is stationary. For s >0, let Z s be distributed as X t+ s X t, t 0. Note that Z s also takes values in N[ f 0g . We are given that, for some >0, P (Z s= 1) = s+o(s ) P (Z s 2) = o(s ) Compute the distribution of X t. (Hint: You may use characteristic functions to deduce the distribution. That is, once you compute the characteristic function, you can directly tell the distribution without having to formally invert it.) 8.(2 points) Let Xhave a probability density function given by f(x ) = c (1+ x2 ) log(1+ x2 ) , where c is a constant to ensure that f(x ) integrates to one. It is rather easy to see that Z1 1 xf (x )dx is not well-de ned. Therefore its mean is not well-de ned.

Let X 1; :::; X nbe mutually independent and have the same law as X. Let A n = X 1+ +X n n :

Show that lim n !1 E e itA n ! 1; 8t 2 R:

Note: Hence from the earlier assignment, we see that weak law can sometimes hold even though the mean is not well-de ned. IE Dept, CUHK [email protected] Page 4 of 4