WINTER 2014/15 TERM 2 STAT 302: ASSIGNMENT 4 Due: Thursday April 2 at 2pm Please remember to include a cover sheet when you submit your assignment. You may hand in your assignment in class or deposit it in the STAT 302 assignment box on the ground oor of the Earth Sciences Building (ESB) by the due time. When answering the questions, writing down the nal answer will not be sucient to receive full marks. Please show all calculations unless otherwise specied. Also dene any events and random variables that you use in your solutions. 1. The joint probability density function of X and Y is given by f (x, y) = (y x)ey , 0 x y < . (a) Find the marginal probability density functions of X and Y . (b) Find the conditional probability density function of X, given Y = y. (c) Are X and Y independent? Briey explain why or why not. (d) Find the conditional cumulative distribution function of X, given Y = y. (e) Evaluate the probability of X > 1 when Y =2. 2. Xi N (0, 1) for i = 1, 2, ... and Y N (0, 100) are all independent of each other. Let Z(t) = t i=1 Xi + Y for t 0. (a) What is the variance of Z(t)? (b) Find the covariance between Z(t1 ) and Z(t2 ) where t1 < t2 . (c) Suppose Y (t1 , t2 ) = Z(t1 )Z(t2 ) t2 where t1 < t2 . What is value of E[Y (t1 , t2 )] when t2 ? 3. Suppose Y = 4X and Z = ln(X) where X U nif (0, 2). Calculate the expectation and variance of Y + Z. You may nd the following formulas useful for your calculation: (ln x)2 dx = x[(ln x)2 2 ln x + 2] + C and 1 x(ln x)dx = x2 (2 ln x 1) + C 4 4. Suppose that at ABC Company there is only one customer representative. Let N Bin(10, 0.6) be the number of customers requiring service in one hour, and Si N (10, 5) be the service time (in minutes) for the ith customer that requires service, independent of other customers. Dene a new random variable T (in minutes) that denotes the total amount of time spent serving customers. (a) Express T in terms of the other random variables dened in the question. (b) Find the expected total amount of time that the customer representative is serving customers. (c) Find the variance of the total amount of time that the customer representative is serving customers. (d) Find the probability that the total service time exceeds 50 minutes. 1 Question number 4: ( Here ) ( ) a) T = Total amount of time spent serving customer = Amount of time spent on 1st customer+2nd customer+...+Nth customer = b) E(T) = E( = c) Var(T) = ( ) ) ( | ) [ , ( | )- , ( | )- , - , - ( ) ( ); As each are independent and have same distribution. , ( | )- * , ( | )-+ ; Using law of total expectation ( ) ( | ) ] * , ( | )-+ { ( | ) ] [ , , ( | )-+ } ( | )- ( ( )) * , - 390 The steps of C is hard so just inform me if you don't understand any step