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PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICAL AND PHYSICAL SCIENCES
SEMESTER 1 EXAMINATIONS, JANUARY/FEBRUARY 2008
MA260 10 – Distributions and Estimation
Time allowed – 2 hours
All questions may be attempted
Marks gained from questions in Section B will be given greater consideration in assessing a first class performance.
Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Statistical Tables will be provided
y
Section A
1 The random variable Y has the probability density function (pdf)
g y( ) = 14 y exp (− y)
for positive y and zero otherwise. (a) Find the mode of this distribution and sketch the pdf. (b) Use the gamma function to find the mean of Y.
[4]
[6]
2 The moment generating function of the random variable X is given by
M X ( )t = ( k −e −^2 t)− 7
(i) Find the value of k. (ii) Deduce and identify the distribution of Y = ½X-7.
[2]
[6]
3 The discrete random variables X and Y have the joint probability mass function P(X=x,Y=y) = c(1+x^2 +2y) for x = -1,0,1 and y = 0,1,2. (a) Find the value of c; (b) Deduce the marginal distribution of Y and the conditional distribution of X given that Y=1. (c) Evaluate P(X^2 =1|Y=1). [8]
4 The continuous random variables X and Y are defined in the region illustrated, viz. {(x,y):02X) and ^2 Y X
E.
[4]
[6]
x
x=y
Section B
8 The discrete random variable X takes the values 0, 1 and 2 with equal probability.
(i) Show that ( 1 ) 1 ( 3 1)
k^ k E ^ X − = + − , stating for what k this result is valid. (ii) Find the moment generating function (mgf) of X and from this
deduce that Y = X–1 has mgf M Y ( )t = 13 ( 1 + e t^ + e−t).
(iii) Use the mgf of Y to deduce the moments about the origin of Y and verify that your results agree with those in (i). [Recall that 0!
u^ n n
e u n
∞
= ∑ for all u ]
[10]
9 U and V have the joint probability density function (^52) / 2 ( , ) 3 v 0 g u v v e u v u
and zero otherwise. If X = U V andY = V^2 find the joint probability density function of X and Y and deduce that (i) the (marginal) density function of X is f x ( ) = (^) x^23 for a range of values (which should be stated); (ii) Y has a chi-squared distribution. Are X and Y independent? (^) [20]
10 The four observations x 1 =+1, x 2 =–3, x 3 =0, x 4 =+4 form a random sample from a continuous distribution that has a uniform distribution over the range (–θ,θ). Quoting standard properties of this distribution where necessary, find (i) the moment estimate of θ; (ii) the maximum likelihood estimate of θ; [10]
11 Based on a random sample of 15 observations from a distribution with parameter θ, a statistician has computed the likelihood function (^152 3) / 15 45
L θ P e Sθ θ
=^ −
where P and S represent the product and sum of the data respectively. Show that the maximum likelihood estimator of θ is unbiased and calculate its standard error. [You may quote without proof that the mean and variance of S are 15θ and 5θ^2 respectively.] A second statistician claims that he can find an alternative estimator of θ that is unbiased and has standard error θ/7. Comment. (^) [14]
12 If^ X 1 ,^ X 2 , … ,X 7 are independent standard Normal variables, find the value c such that
P ( X 1 2 + X 22 + X 32 + X 42 + X 52 < c X ( 62 + X 72 )=0.05) [6]
