RYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS MTH 514 Stochastic Processes Midterm 2 Assignment Last Name (Print):. First Name:. Student Number: Signature:. Date: March, 2010 Due: March 18, in class. Instructions: Professor G. Ord This is an assignment based on your midterm. Read the instructions carefully before starting. Some of you are still having trouble distinguishing between the sample space and the functions(pdf, PMF, CDF) defined on the sample space. In the following, use set notation to indicate a sample space. eg. S X = {x R 0 x 2} denotes the set of all real x in the closed interval [0, 2]. A two dimensional sample space might be S XY = {(x, y) R 2 0 x π, 0 y 1} Notice that the specification of a sample space tells you the dimension you are working in and, for continuous variables, specifies a region over which you will integrate! In this assignment you may help each other, but ultimately the work you submit must be your own! Xerox copies of work are not acceptable. The mark for each question will be a Bernoulli random X { 1 the answer is complete, perfectly legible, and absolutely correct. X = 0 otherwise. Intermediate small steps in a calculation need not be displayed but you need to indicate grammatically that you know what you are doing. For example xe x dx =... = (x + 1)e x + C (IBP) For Instructor s use only. Page M/C 4 5 6 Total Mark would be acceptable. There are extra notes in blue to guide your thinking. If the note requests an answer, provide it. Make sure your answer is easy to see. If an empty box easy is provided for an answer, put it there. Print out this assignment and do the questions on the appropriate pages. The marker will be quick and ruthless so be neat and organized! Good Luck! 1
Stochastic Processes Midterm 2 Exam, Form: A Name: Student Number: TA: Date: Section 1. Multiple Choice [ 3 marks each] 0 x < 0 1. A random variable X has a CDF given by F X (x) = x 0 x < 0.8 The random variable X is: 1 x 0.8 (a) (b) (c) (d) continuous. discrete. mixed. bivariate. (e) None of the above. Sketch the graph of the CDF indicating the sample space. 2. For the random variable X given above, the value of P[0.25 < X 0.8] is closest to: (a) 0.25 (b) 0.70 (c) 0.75 (d) 1.00 (e) 0.55 3. The random variable Z has probability density function (PDF): { 0 z < 0 f Z (z) = 2 2π exp[ z 2 The expected value of Z, E[Z] is: /2] z 0 (a) 2 2π (b) 0 (c) 1 2π (d) 1 (e) None of the above. What is the sample space here? 2
4. Let Y = X 1 be the output of a shifted full-wave rectifier with input voltage X. If X is a random variable with continuous PDF f X (x) then the probability that Y < y for y > 0 is: (a) (b) 1+y f 1 y X(x) dx y+1 f X(x) dx + f y+1 X(x) dx (c) 2 y+1 f 0 X (x) dx (d) f X (x) dx y (e) None of the above. Sketch the region of S X corresponding to Y < y. 5. Let R be the region R = {(X, Y ) 1 X 1 and 1 Y 1} and let (X, Y ) be a pair of random variables that is distributed uniformly in this region. That is f X,Y (x, y) is constant in this region and 0 elsewhere. The probability P[Y X ] is closest to: (a) 1/2 (b) π/4 (c) 3/4 (d) 1/4 (e) 1.00 Sketch S XY and the region corresponding to Y X. 6. The continuous random { variable X is uniform on the interval [0, 1]. A derived random variable Y is 2 0 X 2 0.81 given by Y (X) = The expected value of Y, E[Y ] is closest to: 4 0.81 X 2 1. (a) 0.9 (b) 3.8 (c) 2.6 (d) 2.2 (e) 3.0 Using set notation describe the sample spaces of X and Y. 7. If the bivariate PDF of (X, Y ) is constant over the unit disk X 2 +Y 2 1 then the marginal probability density for X, f X (x) is: (a) f X (x) = 2 1 x 2 π 1 x 1 (b) f X (x) = 1/2 1 x 1 (c) f X (x) = 1 0 x 1 (d) f X (x) = 2 1 x 2 π 0 x 1 (e) None of the above. Using set notation describe the sample spaces of X and Y. 3
8. Suppose X is a normally distributed random variable with mean µ = 16 and variance σ 2 = 4, The probability Pr{X 14} is closest to: (a) 0.33 (b) 0.73 (c) 0.84 (d) 0.27 (e) 0.16 Sketch the PDF and the region X 14. 4
Section 2. Short Answer 9. [7 marks] Two random variables X and Y have joint PDF { c sin(x) exp( y) 0 x π, y 0 f XY (x, y) = 0 otherwise. (a) Sketch the sample space. (b) Assuming c to be appropriately chosen calculate the CDF? (c) Use the CDF to find the value of the constant c? (d) What is P [X 2, 1 Y 3]? (e) What is the marginal PDF for X? (f) Show that X and Y are either dependent or independent, whichever is the case. (g) What is the common name for the marginal PDF f Y (y)? Unit Normal Distribution 5
10. [7 marks] Two independent random variables (X, Y ) have joint PDF { c exp[ x 2 /2] x R, 0 y 2 f XY (x, y) = 0 otherwise (a) Sketch the sample space, find the marginal PDFs and the value of c. (b) What is the common name for the marginal PDF f X (x)? Unit Normal Distribution (c) What is the Variance of X? Unit tribution (d) What is the common name for the marginal PDF f Y (y)? Unit Normal Distribution (e) Find E[X] and E[Y ]. (f) Find P [ 1 x 2]. (g) Find P [0 y x 1] (Hint: Sketch the region and use vertical strips.) 6
11. [8 marks] Two fair 6-sided die are rolled and a pair of random variables (X, Y ) recorded. Y is the absolute value of the difference of the faces so S Y = {0, 1, 2, 3, 4, 5} and X is the sum of the faces. a) Find the joint PMF, P XY (x, y) and fill out the table below with the natural numbers 36 P XY (x, y). Also provide the values of the marginal PMF s (times 36) in the rows/columns labeled. Y \X 2 3 4 5 6 7 8 9 10 11 12 0 1 2 2 3 4 5 For example notice that the pair (X, Y ) = (3, 1) corresponds to the sum of the faces being 3 and the difference 1. This can only happen if the first die is 1 and the second is 2 or vice versa. The probability of this event is then 2/36 giving the second row, third column entry of 2. (b) Show that X and Y are independent/dependent whichever happens to be the case. (c) Find the conditional PMF P X Y =0. (d) Find µ X = E[X] and µ Y = E[Y ]. (e) Let R be the region: R = {(X, Y ) ( X µ X 1) ( Y µ Y 1)}. Indicate this region above and find P [R]. (f) For what value of Y is the conditional PMF P X Y equivalent to the Bernoulli PMF? 7
Figure 1: Normal Tables 8