Question 1
The random variable X is uniformly distributed on (-1, 2) . Derive the pdf of Y = X2.
Question 2 The random variable X has pdf
c (a, )3) x'1 (1 + x)-°-B x > 0 fx (s) = { 0
; x < 0
for appropriate c (a, /3) . Derive the pdf of Y = X/ (1 + X).
Question 3 The independent random variables X and Y have the standard normal distri¬bution. We wish to find the pdf of Z = X/ Y. Show how to do this by creating a 'dummy' variable U = Y and considering the functional relationship
[ tzl 1 = g ( ra
Question 4 The random variables X and Y are continuous, with joint pdf
, , cc' ; 0= 0 ; otherwise.
(a) Find the constant c;
(b) Find the marginal pdfs of X and Y;
(c) Determine E (X), vat (X), E (Y) , vat (Y) , cov (X, Y) and corr (X, Y);
(d) Are X and Y independent? Give reasons.
(e) Calculate E . Hence, using the pdf of Y, find E (X).