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PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICAL AND PHYSICAL SCIENCES
SEMESTER 1 EXAMINATIONS, JANUARY/FEBRUARY 2009
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
Section A
1 In a raffle held at a charity event the three winning ticket numbers were 7, 48 and 23. The tickets sold were numbered 1,2,…,θ, where θ is unknown. Suggest three estimates of θ that might be considered reasonable. Justify your answers. [6]
2 T 1 and T 2 are both unbiased estimators of an unknown parameter θ. They have respective variances 2σ^2 and 3σ^2 and they are known to be independently distributed. Show that S = aT 1 + (1–a)T 2 is an unbiased estimator of θ for all values of the constant a. What value would you recommend for a? Why? What is the efficiency of T 1 relative to the optimal S? [10]
3 The continuous random variable Y has the probability density function given by f (y) = 3/y^4 , y>1 (and zero otherwise). Use the Transformation Theorem to find the probability density function of W = 1/Y 2. [8]
4 Discrete random variables X and Y have the joint distribution given by P(X=x,Y=y) = c (xy + x + 1) for all pairs such that x = 0,1,2 and y = 0,1. No other values are possible. Find the value of c and the conditional distribution of X given that Y=1. Evaluate also E[X] and E[X|Y=1]. [8]
5 The continuous random variables (X,Y) have the joint probability density
function ( , ) 1 ( 1 4 )
f x y = 2 + xy for 0<x<1, 0<y<1 and zero otherwise. Calculate (i) the marginal distribution of X; (ii) the conditional distribution of Y given that X=1/3; (iii) the chance that Y is greater than X. (^) [12]
6 The random variable X has the moment generating function MX(t) = (ket–1)–8. Deduce the value of k and give the mean and variance of X. [8]
Show that the probability density function of L, the largest of the five values is given by
9 10 g l ( ) = 10 l θ for 0<l<θ^ and derive the density function of^ M,^ the median of the five values. Deduce an unbiased estimator of θ.
[9]
[3]
10 The range space of the random pair(X ,Y) is the triangle whose vertices are the three points (0,0), (1,1) and (1,0); their density function over this range is fXY(x,y) = cxy and is zero for all other pairs of values. (i) Calculate P(Y<uX) where u is a constant. For what u is this valid?
(ii) Evaluate Y
X
E .
(iii) Verify that the transformation from (X,Y) to (U,V) where U = YX and V = X^2 is one-to-one, and give the range space of (U,V). (iv) Evaluate the Jacobian of this transformation and deduce the joint distribution of (U,V). (v) Find the marginal distribution of U. How does this relate to your answer in (i)?. (vi) Are U andV independent? Give reasons for your answer
[4]
[3]
[5]
[6]
[4]
[3]
11 Define the moment generating function MX(t) of a random variable X and prove that the moment generating function of Y =aX+b is MY(t) = ebtMX(at) The random variable X has the moment generating function
( ) 1 exp
X 6 2
M t = ^ − t^ ^ − t
,^ t<6.
Find values a and b so that Y has a chi-squared distribution.
[4]
[6]
