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CS 3190: Foundations of Data Analysis

Homework 1: Probability and Bayes’Rule

1. [20 points] Using the probability table below for the random variables X and Y , derive the following values

(a) Pr(X 0)

(b) Pr(X = 0 u Y = 0)

(d) Are X and Y independent? and explain why.

	X = 0	X = 1
Y = 0	1/10	2/10

2. [25 points] An “adventurous” track athlete has the following running routine every morning: He takes a bus to a random stop, then hitches a ride, and then runs all the way home. The bus, described by a random variable B , has four stops where the stops are at a distance of 5, 8, 11, and 12 miles from his house – the ﬁrst three stops have probability 1/6 of occurring. The 12 mile stop has probability 1/2 of occurring. Then the random hitchhiking takes him further from his house a uniformily distributed number of miles on the distances _4 to 5; that is it is represented as a random variable H with pdf described

f (H = x) = ,0(1)/9

if x e [_4, 5]

if x [_4, 5]

Note that a negative distance means that the runner is taken closer to his house. For example, if H = _1, then the runner is taken 1 mile ℃lóser to his home.

What is the expected distance he jogs each morning?

3. [30 points] Consider a data set D with three data points {_1, 0, 1}. Assume the data has Laplacian noise deﬁned with location M and scale 1, so from a model M a data point’s

probability distribution is described by fM (x) = exp(_ |M _ x|). We want to choose M from the space Ω = {_3, _1, 7}. Also assume we have a prior knowledge assumption on the model that Pr(M = _3) = 0.75, Pr(M = _1) = 0.1, and Pr(M = 7) = 0.15. Assuming all data points in D are independent, which model is most likely?

4. [25 points] The Laplace Distribution, indexed by location parameter µ and scale parameter σ has probability density function given by f (x) = exp(_ μ|) for x e R, µ e R and σ > 0. Plot the pdf and cdf of a Laplace random variable with µ = 3 and σ = 1 for values of x is range [_3, 9]. The function scipy .stats .laplace may be useful.