A sample ﬁnal
Assume for the purposes of this exam that I have given you all necessary pmfs and pdfs, and necessary moment
generating functions. (I do not expect you to memorize all of the moment functions. I may ask you to derive
one, so you need to know what it is, and how to derive it for the common distributions.)
1. (3 parts worth 2 marks each)
(a) What is P(B∩(A∪C))
(b) What is P(A∪(B∪C))
(c) Are the two events C∩Band A∩Bindependent?
2. Suppose Xi∼N(µi, σ2
i),i= 1,2, . . . , n. Using whatever means you feel appropriate, derive
the distribution of Pn
i=1 aiXi. Include the parameters of the distribution, and the support of the
density function. (Assume that you have been given the moment generating function of the normal
3. Suppose X∼Exp(θ). Using the cumulative distribution function technique, derive the distribu-
tion of Y=√X
2+ 8. Make sure you include what range of values Ycan take on (i.e. the support
4. Suppose X∼Unif(0,1). Using the transformation method, derive the distribution of Y=ln(−ln(X)).
Make sure you include what range of values Ycan take on (i.e. the support of fY(y)).
5. Suppose X∼Exp(2),Y∼χ2(1),Z∼N(0,1), and X,Y, and Zare independent.
(a) What is the joint distribution of X,Y, and Z?
(b) What is the moment generating function of X+Y+Z? (If you have been given the MGFs
of X,Y, and Z.)
(c) What is E(XY Z)?
(d) What is E(XY Z2)?
6. Prove that the addition rule holds under conditioning. That is, show that
P(A∪B|C) = P(A|C) + P(B|C)−P(A∩B|C).
7. If X∼Gamma(3,4), show that f(x)is increasing in xfor 0< x < 8, decreasing in xfor x > 8,
and has a maximum at x= 8.
8. If X∼Gamma(α, β)where α > 0, β > 0derive E(X3).
9. Derive an expression for V ar(X−Y)in terms of V ar(X),V ar(Y), and Cov(X, Y ).
10. Show that if Xi∼Exp(θ),i= 1, . . . , n, then Pn
i=1 Xi∼Gamma(n, θ).
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