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COM 2004 probability theory

创建日期：2024-04-02 14:36:16

所属分类：统计学statistics

标签： COM 2004

probability theory

COM 2004 probability theory

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%]
(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%]
(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%]
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%]
(ii) Calculate the probability P(X1 = H) in terms of λ. [5%]
(iii) Calculate the probability P(X2 = H) in terms of λ. [5%]
(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%]
(v) Calculate the value of ρ. [5%]
COM 2004 3 TURN OVER

COM 2004
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%]
(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%]
COM 2004 4 CONTINUED
COM 2004
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%]
(ii) Write down the formula for sample variance. Use it to calculate the unbiased
variance estimate for both height and arm span. [10%]
(iii) Write down the formula for sample covariance. Use it to calculate the unbiased
estimate of the covariance between height and arm span. [10%]
(iv) Write down the covariance matrix, Σ. [5%]
(v) Compute the inverse covariance matrix, Σ−1. [15%]
b) Remember that the pdf of a multivariate Gaussian is given by
p(x) =Ce−
1
2
(x−µ)TΣ−1(x−µ)
whereC 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%]
(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%]
COM 2004 5 TURN OVER
COM 2004
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%]
(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.
Using the diagram estimate the proportion of the population who will have an ape
index greater than 10.5? [5%]
(iii) Using the figure above estimate the mean-centred range of ape indexes that
would include 99% of the population. [5%]
COM 2004 6 CONTINUED
COM 2004
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%]
(ii) The mean of class 1 and class 2 are decreased by equal amounts. [5%]
(iii) The prior probability of class 2 is increased. [5%]
(iv) The variance of class 1 and class 2 are increased by equal amounts. [5%]
(v) The variance of class 2 is increased. [5%]
b) Consider a Bayesian classification system based on a pair of 2-D multivariate normal
distributions, p(x|ω1) ∼ N(µ1,Σ1) and p(x|ω2) ∼ N(µ2,Σ2) . The distributions have
the following parameters
µ1 =
(
1
2
)
µ2 =
(
3
5
)
Σ1 = Σ2 =
(
1 0
0 1
)

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