Description
1. Transformation of Normal random variables (Total 5 points)
(a) If π~ππππππ(π, π2) and π = ππ + π, with a > 0, prove that π~ππππππ(ππ + π, (ππ)2). (3 points)
(b) If πand π are i.i.d. standard Normal RVs, show that π + π ~ ππππππ(0, 2). In general, the sum of
any two independent Normal RVs is also a Normal. (2 points)
2. Introduction to Covariance (Total 5 points)
The covariance of two RVs X and Y is defined as: Cov(X,Y) = E[(X – E[X]) (Y – E[Y])] = E[XY] – E[X] E[Y].
Covariance of independent RVs is always zero.
(a) In an experiment, an unbiased/fair coin is flipped 3 times. Let X be the number of heads in the first
two flips and Y be the number of heads in the last two flips. Calculate Cov(X,Y). (2 points)
(b) Let X be a fair 5-sided dice with face values {-5, -2, 0, 2, 5}. Let Y = X2
. Calculate Cov(X,Y). (2 points)
(c) Does a zero covariance imply that the RVs are independent? Justify your answer. (1 point)
3. Inequalities (Total 10 points)
Let X be a non-negative RV with mean π and variance π2, and let t > 0 be some real number.
(a) Prove that πΈ[π] β₯ β« π₯π(π₯)ππ₯ β
π‘ . (3 points)
(b) Using part (a), prove that Pr(π > π‘) β€ πΈ[π]
π‘ (3 points)
(c) Using part (b), prove that Pr(|π β π| β₯ π‘) β€ π2
π‘2 (4 points)
4. Functions of RVs (Total 10 points)
(a) Let π1,π2, β¦ , ππ be π independent exponential random variables with pdfs given by
πππ
(π₯) = πππβπππ₯, π₯ β₯ 0, β π β {1, 2, β¦ , π}. Let π = min (π1,π2, β¦ , ππ).
i. Find the pdf of Z. (3 points)
ii. Find E[Z]. (1 point)
iii. Find Var(Z). (2 points)
(b) Let π and π be two random variables with joint density function:
πππ(π₯, π¦) = οΏ½
2, 0 β€ π₯ β€ π¦ β€ 1
0, ππ‘βπππ€ππ π . Find the pdf of π = ππ. (4 points)
5. Daenerys returns to Kingβs Landing, almost. (Total 10 points)
In an alternate universe of Game of Thrones (or A Song of Ice and Fire, for fans of the books), Daenerys
Targaryen is finally ready to leave Meereen and return to Kingβs Landing. However, she does not know
the way. From Meereen, if she goes East, she will wander around for 20 days in the Shadow Lands and
return back to Meereen. If she goes West from Meereen, she will immediately arrive at the city of
Mantarys. From Mantarys , she can go West by road or South via ship. If she goes South, her ship will get
lost in the Smoking Sea and will be swept back to Meereen after 10 days. However, if she goes West
from Mantarys, she will eventually reach Kingβs Landing in 5 days. Let X denote the time spent by
Daenerys before she reaches Kingβs Landing. Assume that she is equally likely to take either of two paths
whenever presented with a choice and has no memory of prior choices.
(a) What is E[X]? (3 points)
(b) What is Var[X]? (7 points)
(Hint: Be careful with Var[X]. You want to use conditioning.)
6. Stay Away From Stocks! (Total 15 points)
It has always been your dream to own a CMW car, which costs $Y, but unfortunately you only have $X,
with X < Y both being positive integers. To overcome this shortage, you decide to bet on the stock
market and buy shares of a stock for $X. The stock value is known to change as a simple random walk,
i.e., its value either increases or decreases by $1 every day with probability 0.5. Assume that (i) you will
liquidate your stock (convert stock into cash) once your stock reaches the target value of $Y, and (ii) if
your stock value decreases to $0, then you canβt recover (the two stopping criteria). Model the scenario
as a discrete time Markov chain to answer the questions below.
(a) What is the probability that your stock value reaches the high of $π? (4 points)
(b) What is the probability that your stock value reaches low of $0? (3 points)
(c) What is the expected value of your stock at the end? (2 points)
(d) Solve parts (a), (b), and (c) above via simulation (in python). Simulate the stock value as a random
walk. To simulate a random walk, start with an initial value as the current state. Generate a uniform
random variable π’ = π[0, 1], if π’ < π (π is the probability of increase in stock value), then increase
the current state by 1, else decrease it by one. Keep repeating this process until you meet either of
the stopping criteria. To calculate the above-mentioned probability of events and the expected
values, we will go with the frequentist interpretation of probability based on large number of
repeated trials. π(π΄) = ππ΄
π
and πΈ[π] = βππ
π
where ππ΄ is the number of favorable events and π is the
number of trials and ππ is the outcome of π
π‘β trial. So essentially you will be repeating the random
walk π times (π β« 1) and calculate the quantities asked for in 6(a) — 6(c).
Submit your code via the google form link https://forms.gle/uFF4U4Th7YYAhxRn6. For this question,
your code submission should include a python file, a2_6.py. The script should have a function [a,
b, c]ο rand_walk(init_val, final_high, final_low, prob, N) where the
returned values a, b, c, are the answers for 6(a) — 6(c), respectively, and the function
arguments init_val, final_high, final_low, prob and N are initial stock value,
final stock high, low values (stopping criteria), probability of upward movement, and the number of
trials, respectively. For 6(d), also mention the final answers in your hardcopy submission for the
following test cases in the specified format. (6 points)
Test cases
Case # Init_val Final_high Final_low Prob N
1 100 150 0 0.50 10000
2 100 200 0 0.52 10000
3 200 250 0 0.54 10000
Output format
TEST CASE 1 >> (answer for 6(a)) (answer for 6(b)) (answer for 6(c))
TEST CASE 2 >> (answer for 6(a)) (answer for 6(b)) (answer for 6(c))
TEST CASE 3 >> (answer for 6(a)) (answer for 6(b)) (answer for 6(c))
7. Dependence on past 2 states (Total 15 points)
Consider the Clear-Snowy problem from class. However, this time, assume that the weather tomorrow
depends on the weather today AND the weather yesterday. While this does not seem to follow the
Markovian property, you can modify the state space to work around this issue. Use the following
notation and transition probability values:
Pr[ Weather tomorrow is Xi+1, given that weather today is Xi and weather yesterday was Xi-1 ]
= Pr[ Xi+1 | Xi Xi-1 ] (note that each X is either c or s).
Pr[ c | c c ] = 0.9; Pr[ c | c s ] = 0.8; Pr[ c | s c ] = 0.5; Pr[ c | s s ] = 0.1.
(a) Find the eventual (steady-state) Pr[ c c ], Pr[ c s ], Pr[ s c ], and Pr[ s s ]. Show your Markov chain and
the transition probabilities. (7 points)
(b) In steady-state, what is the probability that it will be snowy 3 days from today. (3 points)
(c) Solve the problem of finding the steady state probability via simulation (in python). You need to find
the steady state by raising the transition matrix to a high power (π
= ππ; π β« 1) and then take any
row of the exponentiated matrix (π
[π, βΆ]) as the steady state. For taking power of matrix in python,
you can use np.linalg.matrix_power(matrix, power). After you obtain the steady state distribution,
solve part (b) numerically. (5 points)
Submit your code via the google form link https://forms.gle/uFF4U4Th7YYAhxRn6. For this question,
your code submission should include a python file, a2_7.py. The script should have a function a ο
steady_state_power (transition matrix), where steady_state_power () should have the
implementation of Power method and the return value a is the final steady state. Also, in
the hardcopy submission, you should mention the final steady state you obtained in the
following format:
Steady_State: Power iteration >> [xx, xx, xx, xx]
