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CSE 544 Probability and Statistics for Data Science

Assignment 2: Random Variables

I/We understand and agree to the following:

(a) Academic dishonesty will result in an ‘F’ grade and referral to the Academic Judiciary.

(b) Late submission, beyond the ‘due’ date/time, will result in a score of 0 on this assignment.

(write down the name of all collaborating students on the line below)

1. 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)

2. Inequalities (Total 10 points)

Let X be a non‐negative RV with mean ? and variance ?ଶ, and let t > 0 be some real number.

(a) Prove that ?ሾ?ሿ �?� ሺ?ሻ?? ஶ

௧ . (3 points)

(b) Using part (a), prove that Prሺ??ሻ ாሾሿ

௧ (3 points)

(c) Using part (b), prove that Prሺ|?െ?| ?ሻ ఙమ

௧మ (4 points)

3. Functions of RVs (Total 10 points)

(a) Let ?ଵ, ?ଶ,…,? be ? independent exponential random variables with pdfs given by

?

ሺ?ሻ ൌ ??ିఒ௫, ? 0, ∀ ? ∈ ሼ1, 2, … , ?ሽ. Let ? ൌ min ሺ?ଵ, ?ଶ,…,?ሻ.

i. Find the pdf of Z. (3 points)

ii. Find E[Z]. (1 point)

iii. Find Var(Z). (2 points)

(b) Let ? and ? be two random variables with joint density function:

?ሺ?, ?ሻ ൌ ൜2, 0 ? ? 1

0, ??ℎ?????? . Find the pdf of ? ൌ ??. (4 points)

4. Daenerys returns to King’s 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 King’s 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 King’s Landing in 5 days. Let X denote the time spent by

Daenerys before she reaches King’s 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.)

5. 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 (?ൌ?;? ≫ 1) and then take any

row of the exponentiated matrix (?ሾ?, ∶ሿ) 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 along with your solution as part of the zip/tar file on BB. Name your python file

a2_5.py. The script should have a function a 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]

6. Multivariate Normal (Total 10 points)

A random vector X = (X1, …, Xk) is said to have a Multivariate Normal distribution if every linear

combination of Xj has a Normal distribution. That is, we require t1X1 + t2X2 + … + tkXK to have a Normal

distribution for any real values of t1, …, tk. As a special case, we consider t1X1 + t2X2 + … + tkXK to be a

degenerate normal distribution with variance 0 if t1X1 + t2X2 + … + tkXK is a constant (such as when all tj’s

are 0).

(a) If X = (X1, …, Xk) is a Multivariate Normal, show that the distribution of any Xj is Normal. (1 point)

(b) It is possible to have normally distributed random variables X1, …, Xk such that (X1, …, Xk) is not

Multivariate Normal: Let X = Normal(0, 1) and S = 1 with probability ½ and ‐1 with probability ½.

Then Y = SX is normal due to the symmetry of the Standard Normal. In this case show that (X, Y) is

not a Multivariate Normal. (2 points)

(c) Let Z, W be i.i.d Normal(0, 1) random variables. Show that (Z, W) and (Z + 2W, 3Z + 5W) are

Multivariate Normals. (2 points)

(d) If X = (X1, …, Xn) and Y = (Y1, …, Ym) are Multivariate Normal vectors with X independent of Y, then

show that the concatenated vector W = (X1, …, Xn, Y1, …, Ym) is also a Multivariate Normal. (2 points)

(e) Fact 1 (Uncorrelated implies independence): If X is a Multivariate Normal that can be written as X =

(X1, X2), where X1 and X2 are subvectors, and every component of X1 is uncorrelated with every

component of X2, then X1 and X2 are independent.

Fact 2 (Property of Covariance): For any random variables X, Y, W, and V, we have:

Cov(aX + bY, cW + dV) = ac Cov(X, W) + ad Cov(X, V) + bc Cov(Y, W) + bd Cov(Y, V).

Let X, Y be i.i.d. standard Normals. Use Fact 1 and Fact 2 to show that (X+Y, X‐Y) is a Multivariate

Normal. (3 points)

7. Pokémon Go fanatic (Total 10 points)

Let us assume there are only n distinct types of Pokémons to capture in the entire Pokémon world,

though there is an infinite supply of each type. Every day, you capture exactly one Pokémon. The

Pokémon that you capture could be any one of the n types of Pokémons with equal probability. Your

goal is to capture at least one Pokémon of all n distinct types. Let X denote the number of days needed

to complete your goal.

(a) What is E[X]? (5 points)

(b) What is Var[X]? (5 points)

We do not need closed‐forms here for parts (a) and (b).