# CS 4964: Math for Data Homework 1: Probability and Bayes’ Rule

\$35.00

## Description

1. [15 points] For the random variables X and Y , derive the following values
(a) Pr(X = 1)
(b) Pr(X = 2 ∩ Y = 1)
(c) Pr(X = 3 | Y = 2)
X = 1 X = 2 X = 3
Y = 1 0.1 0.05 0.2
Y = 2 0.05 0.25 0.35
2. [20 points] Consider rolling two fair die D1 and D2; each has a probability space of Ω =
{1, 2, 3, 4, 5, 6} which each value equally likely. What is the probability that D1 has a larger
value than D2? What is the expected value of the sum of the two die?
3. [10 points] Let X be a random variable with a uniform distribution over [0, 2]; its pdf is
described
f(X = x) = (
1/2 if x ∈ [0, 2]
0 if x /∈ [0, 2].
What is the probability that f(X = 1)?
4. [30 points] Consider a data set D with three data points {−1, 7, 4}. We want to find a
model for M from a restricted sample space Ω = {1, 3, 5}. Assume the data has Laplacian
noise defined, so from a model m a data point’s probability distribution is described f(x) =
1
6
exp(−|m − x|/3). Also assume we have an assumption on the models so that Pr(M = 1) =
0.4, Pr(M = 3) = 0.3, and Pr(M = 5) = 0.3. Assuming all data points in D are independent,
which model is most likely.
5. [25 points] Use python to plot the pdf and cdf of the Laplace distribution (f(x) =
1
2
exp(−|x|)) for values of x in the range [−3, 3]. The function scipy.stats.laplace may be
useful.