MATH20722
Forty minutes
THE UNIVERSITY OF MANCHESTER
FOUNDATIONS OF MODERN PROBABILITY
19 March 2020
10:05 – 10:45
Answer BOTH questions.
(The two questions are worth 20% of the final mark.)
Electronic calculators are not permitted.
c© The University of Manchester
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MATH20722
Answer BOTH questions
Question 1.
(1.1) State the definition of a probability space (Ω,F ,P) . [3 marks]
(1.2) Show that P(lim inf n→∞An) ≤ lim inf n→∞ P(An) for An ∈ F
with n ≥ 1 . [3 marks]
(1.3) Let P1 and P2 be two probability measures on (R,B(R)) such
that P1((−∞, a)) = P2((−∞, a)) for all a ∈ Q . Stating clearly
any result to which you appeal, show that P1 = P2 on B(R) . [4 marks]
Question 2.
(2.1) State the definition of a random variable. [3 marks]
(2.2) Show that if Xn is a random variable for n ≥ 1 then
lim supn→∞Xn is a random variable. [3 marks]
(2.3) Show that if X is a random variable then sin(X) + cos(X)
is a random variable. [4 marks]
END OF EXAMINATION PAPER
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