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DATA DRIVEN COMPUTING

创建日期：2024-04-24 17:36:50

所属分类：经济Economics

标签： COM 2004

DATA DRIVEN COMPUTING

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DATA DRIVEN COMPUTING
Answer THREE questions.
All questions carry equal weight. Figures in square brackets indicate the percentage of avail-
able marks allocated to each part of a question.
COM 2004 1

1. This question concerns probability theory.
a) The discrete random variable X represents the outcome of a biased coin toss. X has
the probability distribution given in the table below,
x H T
P(X = x) θ 1−θ
where H represents a head and T represents a tail.
(i) Write an expression in terms of θ for the probability of observing the sequence
H, T, H, H. [5%]
ANSWER:
θ× (1−θ)×θ×θ = θ3(1−θ)
(ii) A sequence of coin tosses is observed that happens to contain NH heads and
NT tails. Write an expression in terms of θ for the probability of observing this
specific sequence. [5%]
ANSWER:
θNH (1−θ)NT
(iii) Show that having observed a sequence of coin tosses containing NH heads and
NT tails, the maximum likelihood estimate of the parameter θ is given by
NH
NH +NT
[20%]
ANSWER:
Write down the expression for p(x) in terms of the parameter θ,
P(x;θ) = θNH (1−θ)NT
Find maximum by differentiating w.r.t. the parameter θ and setting to 0,
dP(x;θ)
dθ
= NHθ
NH−1(1−θ)NT −θNH NT (1−θ)
NT−1 = 0
Solve for θ,
NHθ
NH−1(1−θ)NT −θNH NT (1−θ)
NT−1 = 0
NH(1−θ)−θNT = 0
NH = NHθ+NT θ
θ =
NH
NH +NT

b) The discrete random variables X1 and X2 represent the outcome of a pair of independent
but biased coin tosses. Their joint distribution P(X1,X2) is given by the probabilities in
the table below,
X1 = H X1 = T
X2 = H λ 3λ
X2 = T 2λ ρ
(i) Write down the probability P(X1 = H,X2 = H). [5%]
ANSWER:
λ
(ii) Calculate the probability P(X1 = H) in terms of λ. [5%]
ANSWER:
λ+2λ = 3λ
(iii) Calculate the probability P(X2 = H) in terms of λ. [5%]
ANSWER:
λ+3λ = 4λ
(iv) Given that the coin tosses are independent and that λ is greater than 0, use your
previous answers to calculate the value of λ. [15%]
ANSWER:
Independence implies that
P(X1 = H,X2 = H) = P(X1 = H)×P(X2 = H)
So
λ = 3λ×4λ
λ =
1
12
COM 2004 3
UNIVERSITY OF SHEFFIELD COM 2004
(v) Calculate the value of ρ. [5%]
ANSWER:
Also entries in the table must sum to one so
λ+3λ+2λ+ρ = 1
Therefore
ρ =
1
2

c) Consider the distribution sketched int the figure below.
0 1
2λ
λ
b
p(x)
x
p(x) =


2λ if 0 <= x < b
λ if b <= x <= 1
0 otherwise
(i) Write an expression for λ in terms of the parameter b. [15%]
ANSWER:
2λb+(1−b)λ = 1
λ =
1
1+b
(ii) Two independent samples, x1 and x2, are observed. x1 has the value 0.25 and
x2 has the value 0.75. Sketch p(x1,x2;b) as a function of b as b varies between
0 and 1. Using your sketch, calculate the maximum likelihood estimate of the
parameter b given the observed samples. [20%]
ANSWER:
• For b in the range 0 to 0.25, p(x1,x2) = λ×λ =
1
(1+b)2
, i.e., 1 to 16/25
• For b in the range 0.25 to 0.75, p(x1,x2) = λ× 2λ =
2
(1+b)2
, i.e., 32/25 to
32/49
• For b in the range 0.75 to 1.0, p(x1,x2) = 2λ×2λ =
4
(1+b)2
, 64/49 to 1
The p(x1,x2) has a maximum value of 64/49 when b = 0.75.

2. This question concerns the multivariate normal distribution.
a) Consider the data in the following table showing the height (x1) and arm span (x2) of a
sample of 8 adults.
x1 151.1 152.4 152.9 156.8 161.8 158.6 157.4 158.8
x2 154.5 162.2 151.5 158.2 165.3 165.6 159.8 162.0
The joint distribution of the two variables is to be modeled using a multivariate Gaussian
with mean vector, µ and covariance matrix, Σ.
(i) Calculate an appropriate value for the mean vector, µ. [5%]
ANSWER:
(156.2,159.9)
(ii) Write down the formula for sample variance. Use it to calculate the unbiased
variance estimate for both height and arm span. [10%]
ANSWER:
Var(x) =
1
N−1
N
∑
i=1
(xi−µx)
2
The variance of height and arm space are 13.9 and 24.9, respectively.
(iii) Write down the formula for sample covariance. Use it to calculate the unbiased
estimate of the covariance between height and arm span. [10%]
ANSWER:
Cov(x,y) =
1
N−1
N
∑
i=1
(xi−µx)(yi−µy)
The covariance between height and arm span is 13.5.
(iv) Write down the covariance matrix, Σ. [5%]
ANSWER:
(
13.9 13.5
13.5 24.9
)

(v) Compute the inverse covariance matrix, Σ−1. [15%]
ANSWER:
(
0.154 −0.0840
−0.0840 0.0860
)
b) Remember that the pdf of a multivariate Gaussian is given by
p(x) =Ce−
1
2 (x−µ)
T Σ−1(x−µ)
where C is a scaling constant that does not depend on x.
Using the answer to 2 (a) and the equation above, answer the following questions.
(i) Who should be considered more unusual:
• Ginny who is 162.1 cm tall and has arms 164.2 cm long, or
• Cho who is 156.0 cm tall and has arms 153.1 cm long?
Show your reasoning. [20%]
ANSWER:
Compute −(x− µ)T Σ−1(x− µ) for each person and see which is smaller. For
Ginny it is -2.67 and for Cho it is -3.71. Therefore Cho’s proportions are more
unusual.
(ii) A large sample of women is taken and it is found that 120 have measurements
similar to those of Ginny. How many women in the same sample would be ex-
pected to have measurements similar to those of Cho? [15%]
ANSWER:
The samples should be in the ratio
exp(−(x1−µ)
T Σ−1(x1−µ)) : exp(−(x2−µ)
T Σ−1(x2−µ))
Plugging in the numbers from the previous section
120∗ exp(−3.71)/exp(−2.67) = 42
COM 2004 7

c) A person’s ‘ape index’ is defined as their arm span minus their height.
(i) Use the data in 2 (a) to estimate a mean and variance for ape index. [10%]
ANSWER:
µ = 3.66
σ2 = 11.6
σ = 3.41
(ii) The figure below shows a standard normal distribution, i.e., X ∼ N(0,1). The
percentages indicate the proportion of the total area under the curve for each
segment.
−3 −2 −1 0 1 2 3
19.1%19.1%
15.0%15.0%
9.2%9.2%
4.4%4.4%
1.7%1.7%
0.5%0.5%
Using the diagram estimate the proportion of the population who will have an ape
index greater than 10.5? [5%]
ANSWER:
10.5 is 2 σ greater than the mean. Therefore about 2.2% of the population will
be expected to have an ape index of 10.5 or greater.
(iii) Using the figure above estimate the mean-centred range of ape indexes that
would include 99% of the population. [5%]
ANSWER:
This would be about + or - 2.5 sigma about the mean. So
• smallest: 3.66−3.41×2.5 =−4.87
• biggest: 3.66+3.41×2.5 = 12.19

i.e., the range [−4.87,12.19].

3. This question concerns classifiers.
a) Consider a Bayesian classification system based on a pair of univariate normal distribu-
tions. The distributions have equal variance and equal priors. The mean of class 1 is
less than the mean of class 2. For each case below say whether the decision threshold
increases, decreases, remains unchanged or can move in either direction.
(i) The mean of class 2 is increased. [5%]

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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