stat

STAT2001 Assignment 5

Do all 6 questions. Deadline for this assignment is 28th November. Please submit your work on Blackboard (as a single pdf file). Q1. (18 marks) The random variables X and Y have the joint probability density function (p.d.f.):

.10 ,),( <<<= + yxceyxf yx (a) Find the constant c. (b) Are X and Y independent? (c) Find the marginal p.d.f. of Y. Q2. (18 marks) X and Y have the following joint p.d.f.:

.10 ,10 ,3),( 2 <<<<= yxxyxf (a) Find the conditional p.d.f. , conditional mean and conditional variance of X given Y=0.3.

(b) Find P(X+Y<0.5).

Q3 (12 marks) Let X has a uniform distribution on the interval (1,2). Give that X=x, Y has a uniform distribution on the interval (0,x). Find the marginal p.d.f. of Y. Q4. (12 marks) Suppose that X follows Exponential distribution with parameter θ=2. And Y follows Gamma distribution with parameters α=5, θ=2. Assume that X and Y are independent. Find the probability P(Z>21) where Z=X+Y. Q5. (15 marks) Consider a bivariate p.d.f. for random variables X and Y:

.10 ,8),( <<<= yxxyyxf

Find the correlation coefficient of X and Y.

Q6. (25 marks) Let X and Y have a Bivariate Normal distribution with parameters 𝜇𝜇𝑋𝑋 = 2, 𝜇𝜇𝑌𝑌 = 4,𝜎𝜎𝑋𝑋2 = 25,𝜎𝜎𝑌𝑌2 = 9 𝑎𝑎𝑎𝑎𝑎𝑎 𝜌𝜌 = 1

3 . Compute

(a) 𝑃𝑃(−1 < 𝑋𝑋 < 1)

(b) 𝑃𝑃(−1 < 𝑋𝑋 < 1|𝑌𝑌 = 5)

(c) 𝑃𝑃(2 < 𝑌𝑌 < 5)

(d) 𝑃𝑃(2 < 𝑌𝑌 < 5|𝑋𝑋 = 1)

(e) 𝑃𝑃(𝑋𝑋 < 𝑌𝑌)

(Remark: I do not include numerical examples about Bivariate Normal in Lecture Notes. You may take a look Example 4.5-2 on P.165 of the textbook for reference.)

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