homework | assignment代做 – CMPT 413/713 Natural language processing

CMPT 413/713 Natural language processing

homework | assignment代做 – , 这是值得参考的assignment代写的题目

homework 0, Fall 2021

Due: Thursday Sep 16th, 2021
``````InstructionsAnswers are to be submitted by 11:59pm of the due date as a PDF file through Canvas.
to be done individually.
``````

1 Derivatives (10pt)

``````Provide the derivatives with respect toxfor each of the following (assume log is the natural logarithm):
``````
``````(a)f(x) =^20 x. 5 +1x 2 (2pt)
``````

(b)f(x) = (ex+ 1)^2 (2pt)

``````(c) f(x) = log (cx^2 +x) (2pt)
``````

(d)f(x) =e(xa) 2 (2pt)

``````(e) f(x) = 1 ^1 ex(2pt)
``````

2 Linear Algebra (6pt)

``````You are given two matrices, please compute their product. Please specify invalid if the two matrices
cannot be multiplied.
``````
``````(a)
``````
]
``````(2pt)
``````

(b)

]
``````(2pt)
``````
``````(c)
``````
]>
``````(2pt)
``````

3 Linear Algebra (5pt)

``````Suppose thatxis a column vector of lengthN(xRN^1 ), andWis matrix of dimensionDrows byN
columns (WRDN). Given that
y=Wx
``````
``````(a) What is the dimension ofy? (2pt)
``````

(b) Suppose we are interested in the derivative ofywith respect tox. How many values does this derivative contain? UseN, Dto denote the final result. (2pt)

``````(c) LetWi,jbe the (i, j)th element ofW. What would beyx^13? (1pt)
``````

4 Linear Algebra (5pt)

``````(a) LetM=
``````
``````xz z
x^2 z ex
exz 1
``````
``````, what is the derivative ofMwith respect tox? (3pt)
``````

(b) What is the rank ofMifx=z= 1? (2pt)

5 Cosine similarity (6pt)

``````Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space. It is
defined as follows:
``````
``````simcos(a,b) =
ab
ab
``````
``````(a) Givena= (2, 2 ,1) andb= (0, 1 ,3), what is the cosine similarity of the two vectorsaandb? (2pt)
``````

(b) What is the minimum and maximum value the cosine similarity can take? (2pt)

``````(c) Given an example of when the cosine similarity is maximized but the Euclidean distance between the
two vectors is greater than zero. (2pt)
``````

6 Probability (5pt)

``````Anna has a weighted coin that has a probabilityphof landing up head, and probabilityptof landing up
tails.
``````
``````(a) Ifph= 0.6, what ispt? (1pt)
``````

(b) Assuming that the flips are independent, if she flips the coin 10 times, what is the probability that she get 10 heads in a row? (2pt)

``````(c) Assuming that the flips are independent, if she flips the coin 10 times, what is the probability that
she get 7 heads and 3 tails? (2pt)
``````

7 Probability (8pt)

``````Twenty percent (20%) of the population has a genetic defect that can lead to a specific disease in later
life. Of the people who have this defect, forty percent (40%) will develop the disease. A test is developed
for detecting the genetic defect. The test can detect the defect ninety percent (90%) of the time and gives
a false positive 5% of the time.
``````
``````(a) What is the probability that Joe (a random person) tests negative for the defect? (3pt)
``````

(b) Joe just got the happy news that the test came back negative; what is the probability that Joe will develop the disease in the future? (5pt)

8 Probability (10pt)

``````LetXbe a continuous random variable with p.d.f.
``````
``````fX=
``````
{
``````2 ex for 1< x < 2
0 otherwise
``````
``````Compute the expectation and variance,E(X) and Var(X).
``````

9 Information Theory (10pt)

``````Suppose there are two distributionsPandQ, and three valuesx 1 ,x 2 , andx 3.
``````
``````x 1 x 2 x 3
distribution P 0.10 0.80 0.
distribution Q 0.25 0.40 0.
``````
``````(a) Compute the entropy ofPand entropy ofQ(4pt)
``````

(b) Calculate the cross-entropyH(P, Q) (4pt)

``````(c) Given arbitrary distributionsP andQ, when is the cross-entropyH(P, Q) minimized? When is it
maximized? (2pt)
``````

10 Information Theory (10pt)

``````The joint distribution of these two random variablesX,Y is as follows:
``````
``````x=a x=b x=c x=d
y=a^1416116118
y=b 0 163 161 0
y=c 0 321 161 321
y=d 321 321 161 0
``````
``````(a) What is the mutual informationI(X;Y) between the two random variablesXandY in bits? (5pt)
``````

(b) What is the range of the mutual information between any two random variables? When is it mini- mized? (5pt)