MTH2222 – Mathematics of Uncertainty 1
MTH2222 – Sem 2, 2016
Assignment 3
Due Date: 5.00 pm on Monday 10 October 2016
Assignments are to be put in the MTH2222 box on the ground floor of Building 28.
1. Let X have the Pareto distribution with shape parameter 1,
fX(x) = x
−2, x > 1.
Find the PDF and CDF of Y = ln(X − 1).
2. Let X be a random variable with PDF
f(x) = c|x|α−1e−λ|x|,
where α > 0 and λ > 0.
(a) Find c.
(b) Identify the conditional distribution of X given X > 0.
(c) Compute the nth moment of X.
(d) Obtain the MGF of X.
3. (a) Let X be a random variable with PDF
f(x) = 1− |x|, |x| ≤ 1.
Obtain the MGF of X.
(b) Find the PDF of a random variable Y with MGF
MY (t) =
(
et − 1
t
)2
,
for t #= 0, and MY (0) = 1.
Assignment due May 26 by 5 pm (via Moodle)
20 pts
5pts
10 pts
1
Deadli e: Tuesday 24 MaylTE⑨B☆BB÷sBBBH@EB
MTH2222 – Mathematics of Uncertainty 2
4. Let the pair (X, Y ) have joint PDF
fX,Y (x, y) = c, x
2 + y2 ≤ 1.
(a) Find c and the marginal PDFs of X and Y .
(b) What are the means ofX and Y ? No calculations are needed, only a brief explanation
is required.
(c) Find the conditional PDF of Y given X = x and deduce E[Y |X = x].
(d) Obtain E[XY ] and compare it to E[X]E[Y ].
(e) Are X and Y independent? Explain.
(f) Obtain var(Y ) without resorting to integration. Hint: Use the fact that var(X) =
var(Y ) and (c).
5. Let U and R be independent continuous random variables taking values in [0, 1]. We
assume that R is uniform and that U has PDF f , CDF F and mean µ. Define the
random variable V sa follows
V =
{
R + (1−R)U if R ≤ 1/2
RU if R > 1/2
(a) Obtain the mean of V in terms of µ.
(b) Obtain an integral expression of the CDF G of V in terms of F .
(c) Deduce an integral expression of the PDF g of V in terms of f .
Hint: You may assume that in this case
d
dt
∫ b
a
φ(t, r)dr =
∫ b
a
∂
∂t
φ(t, r)dr.
(d) Suppose that
f(u) = 6u(1− u), 0 < u < 1.
Compute the mean of V .
Optional. Obtain g.
30 pts
20 pts

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