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SEFYDLIAD MATHEMATEG A FFISEG
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 1 EXAMINATIONS, JANUARY 2010
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.
MA26010: Distributions and Estimation January 2010 Page 2 of 4
Section A
1 A random sample of five observations is taken from a continuous distribution where all values between 0 and θ are equally likely. Suggest reasons why 60 might be a reasonable estimate for the parameter θ if (i) the average of the five observations is 30; (ii) the largest of the five observations is 50. [6]
2 The continuous random variable Y has the probability density function f (y) = 2/y^3 , y>1 (and zero otherwise). (a) Show that P(lnY > z) = e–2z^ ; deduce and identify the distribution of Z = lnY. (b) Use the transformation theorem to find the probability density function of W = 1/Y 2. Identify this distribution. [14]
3 The bivariate discrete pair has the joint probability mass function x 1 2 3
y
1 0.1 2 p p (i) Find the value of p and show that E[X]=2.1. (ii) Give the conditional distribution of X given that Y=1. (iii) How do you know that X and Y are not independent? Are they positively or negatively correlated? [12]
4 The random variables X and Y have the joint probability density function f (x,y) = x + 2y 3 , 0<x<1, 0<y< Find (i) the marginal density of Y; (ii) the conditional density of X given that Y = 2/3; (iii) E[XY ]. [12]
5 X and Y are independent random variables having probability density
functions ( ) 2 0 1 and ( ) 3 (1 )^2 0 X Y 8 f x = x < x < f y = + y < y<. Write down the joint pdf of X and Y and hence evaluate P(X<Y).
[8]
MA26010: Distributions and Estimation January 2010 Page 4 of 4
9 The continuous bivariate pair (X,Y) has the joint probability density function f (x,y) = 8xy over the range space
RXY = { ( x y , ): x > 0, y > 0, x 2 + y^2 < 1 }
U and V are defined by U = X 2 + Y 2 and V = X 2 – Y 2. (i) Verify that this is a one-to-one transformation. (ii) Find the Jacobian of the transformation. (iii) Deduce that the joint density function of (U,V) is constant over its range space. (iv) Show that the range space of (U,V) is in fact
RUV = { ( u v, ): −u < v < +u , 0 < u< 1 }
and check that the area of this range space is as you would expect in the light of (iii).
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[18]
10 Define the moment generating function of a random variable. Prove that the moment generating function of the random variable X having probability density function f x ( ) = 5 e −^5 x for x>0 (and zero
otherwise) is
1 ( ) 1 X 5 M t = ^ −^ t^ − . For what^ t^ is this valid? Use this mgf to find the mean and standard deviation of X. What is the mgf of Y=2X+1? [12]
11 The five observations +2 –3 –4 +2 – comprise a random sample from a U(–θ,+θ) distribution, Calculate (i) the moment estimate, (ii) the maximum likelihood estimate of θ. [10]
