[SOLVED] Homework 1 cs 4650/7650 natural language understanding
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operations unless specified otherwise):
H(X) = −
X
x∈X
P(x) log P(x)
(a) Compute the entropy of the distribution P(x) = Multinoulli([0.2, 0.3, 0.5]). [3
pts]
(b) Compute the entropy of the uniform distribution P(x) = 1
m
∀x ∈ [1, m]. [3 pts]
(c) Consider the entropy of the joint distribution P(X, Y):
H(X, Y ) = −
X
x∈X
X
y∈Y
P(x, y) log P(x, y)
How does this entropy relate to H(X) and H(Y), (i.e. the entropies of the marginal
distributions) when X and Y are independent? [4 pts]
Solution.
Solution goes here. 3. You are investigating articles from the New York Times and from Buzzfeed. Some of
the articles contain fake news, while others contain real news (assume that there are
only two types of news).
Note: for the following questions, write your answer using up to 3 significant figures.
(a) Fake news only accounts for 5% of all articles in all newspapers. However, it
is known that 30% of all fake news comes from Buzzfeed. In addition, Buzzfeed
generates 25% of all news articles. What is the probability that a randomly chosen
Buzzfeed article is fake news? [3 pts](b) Suppose that 15% of all fake news comes from the New York Times (NYT).
Furthermore, suppose that 60% of all real news comes from the NYT. Under all
assumptions so far, what is the probability that a randomly chosen NYT article
is fake news? [3 pts](c) Mike is an active reader of the New York Times: Mike reads 80% of all NYT
articles. However, he also has a suspicion that the NYT is a bad publisher, and
he believes that 25% of all NYT articles are fake news. Furthermore, the NYT
generates 30% of all news articles. Under all assumptions so far, what is the
probability that a randomly chosen article (from all newspapers) will be from the
NYT, will be read by Mike and will be believed to be fake news? [4 pts]
Solution.
Solution goes here. 4. Suppose we have a probability density function (pdf) defined as:
f(x, y) = (
C(x
2 + 2y), 0 < x < 1 and 0 < y < 1,
0, otherwise.
(a) Find the value of C. [2pts]
(b) Find the marginal distribution of X and Y . [4pts]
(c) Find the joint cumulative density function (cdf) of X and Y . [4pts]
Solution.
Solution goes here. 5. [Graduate Students Only] A 2-D Gaussian distribution is defined as:
G(x, y) = 1
2πσ2
exp
−
x
2 + y
2
2σ
2
Compute the following integral:
Z ∞
−∞
Z ∞
−∞
G(x, y) (5x
2
y
2 + 3xy + 1) dx dy
Hint: Think in terms of the properties of probability distribution functions. [5 pts]
Solution.
Solution goes here.