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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.