## 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.