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LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 Hours
Math 313: Probability and Measure
You should answer ALL Section A questions and TWO Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you gain there is capped at 40.
SECTION A
A1. Let A be a σ−algebra on a sample space Ω. (i) State the conditions that P must satisfy if P defines a probability measure on Ω. (ii) Suppose that P is a probability measure and that B is an event such that P(B) > 0. Show that Q(A) = P( PA(^ ∩B^ )B ) (A ∈ A) also defines a probability measure on Ω. (iii) Show that Q(A) = P(A) if and only if A and B are independent events with respect to P. [14]
A2. Let X be a random variable with probability density function
p(x) =
e−x, x > 0 0 , x ≤ 0 ,
and let Y be a random variable with probability density function
q(x) = π^1 1 +^1 x 2 (x ∈ R).
Calculate the expectation of X, and show that Y does not have an expectation. [10]
please turn over
SECTION N continued
A3. Let X be a random variable on (Ω, P) such that EX = μ and EX^2 = K; let X 1 ,... , Xn be mutually independent copies of X. (i) Calculate the expectation and variance of
Zn = X^1 +^ X^2 + n...^ +^ Xn.
(ii) Deduce that Zn → μ in mean square as n → ∞. (iii) State Chebyshev’s inequality, and deduce that Zn → μ in probability as n → ∞. [14]
A4. Let X be a random variable with probability mass function
P[X = k] = (1 − p)pk^ (k = 0, 1 , 2 ,.. .)
for some 0 < p < 1, let Y be an independent copy of X, and let Z = max(X, Y ). (i) Calculate the cumulative distribution functions of X and of Z. (ii) Find the probability mass function of Z. (iii) Give an example of a natural phenomenon for which X and Z will describe quantities that one can observe. [12]
please turn over
SECTION B continued
B2. Let X be a Poisson random variable on (Ω, P) with
P[X = k] = e
−θθk k! (k^ = 0,^1 ,^2 ,.. .). (i) Calculate the characteristic function of X, and hence or otherwise calculate the expec- tation and the variance of X. [14] (ii) Let X 1 ,... , Xn be mutually independent copies of X on (Ω, P). Find the distribution of
Sn = X 1 + X 2 +... + Xn. [6]
(iii) Derive an expression for P[Sn > nθ + t] for t > 0, but do not simplify your answer. [6] (iv) With reference to the axioms for the Poisson process, or to some relevant example, explain why your result in (ii) is reasonable. [4]
B3. Let Z be a standard normal random variable with probability density function
p(z) = e
−z^2 / 2 √ 2 π (z ∈ R).
(i) Calculate the probability density function of X, where X = Z^2. [6] (ii) Let X 1 , X 2 , X 3 , X 4 be mutually independent copies of X. By considering characteristic functions, show that the probability density of V = X 1 + X 2 + X 3 + X 4 is equal to
q(x) =
{ (^) x 4 e−x/^2 ,^ x >^ 0; 0 , x ≤ 0.
[14]
(iii) Calculate the probability density function of W =
V. [6]
(iv) Determine the value of x that makes q(x) largest. [4]
end of exam
