MTH2222 – Mathematics of Uncertainty Assignment 3
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MTH2222 – Mathematics of Uncertainty
Assignment 3
Due on May 26th by 11.55 pm (submit via moodle)
1. Let X have the Pareto distribution with shape parameter 1,
Find the PDF and CDF of Y = ln(X − 1).
2. Let X be a random variable with PDF
(a) Find c.
(b) Identify the conditional distribution of X given X > 2.
(c) Compute the E[X]
(d) Obtain the MGF of X.(Express your answer in terms of CDF of standard normal.)
3. (a) Let X be a random variable with PDF
Obtain the MGF of X.
(b) Find the PDF of a random variable Y with MGF
for t ≠ 0, and MY (0) = 1.
4. Let the pair (X, Y ) have joint PDF
(a) Find c and the marginal PDFs of X and Y .
(b) What are the means of X 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. Defifine the random variable V sa follows
(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) Suppose that
Compute the mean of V .
Optional. Obtain g.
2023-06-10