IE300 BL1 - Analysis of Data Assignment #5 Due Date: October 16, 2015 INSTRUCTIONS: This assignment must be turned in during the first ten minutes of lecture (11:00-11:10AM in 103 Transportation) on the due date. No late submissions will be accepted. As stated in the syllabus, this assignment may be submitted in groups of up to three students (see the syllabus for guidelines and restrictions). Discussion of each of the questions is strongly encouraged among your group. Include your group number and the names of all group members on the top of your assignment. Multi-page submissions must be stapled. Present and explain your solutions clearly. QUESTIONS: 1.) Consider continuous bivariate random variables, X and Y, with the following joint PDF (c is a constant): fXY(x, y) = x - cy for 1 x 2 and 0 y 1 0 otherwise (a) (b) (c) (d) (e) (f) (g) (h) (i) 2.) Find the value of c for which this is a valid PDF Compute P(X 1.25, Y 0.75) Find fX(x), the marginal distribution of X Compute P(1.3 < X < 2.6) Compute E[X] Compute V[X] Find f Y|X = x (y), the conditional distribution of Y given that X = x Find E[Y | X = 1.5] Find P(Y > 0.75 | X = 1.5) Consider discrete bivariate random variables, X and Y, with the joint PMF given in the following table. X fXY(x,y) 2 3 6 1 0.20 0.05 0.05 Y 4 0.05 0.15 0.03 5 0.12 0.10 0.25 (a) (b) (c) (d) (e) (f) (g) Compute P(X Y) Find fX(x), the marginal distribution of X Find fY(y), the marginal distribution of Y Find fY|X = 3 (y), the conditional distribution of Y given that X = 3 Compute E[X] and V[X] Compute E[Y | X = 3] and V[Y | X = 3] Compute Cov(X, Y) and corr(X, Y) 3.) Suppose that desks shipped from a local furniture warehouse are of premium quality with probability 0.40, good quality with probability 0.25, and poor quality with probability 0.35. Each desk is independent of the other desks. Suppose that 18 desks are shipped in a single shift at the warehouse. Of these 18 desks, let X, Y, and Z count the number of desks of premium quality, good quality, and poor quality, respectively. Compute the following. (a) P(X = 12, Y = 4, Z = 2) (b) P(Y = 8) (c) P(X = 10 | Y = 6) (d) P(X 10 | Z = 7) [HINT: If you know the values of two random variables, what do you know about the third variable?] 4.) Consider independent normal random variables, X and Y, with E[X] = 10, V[X] = 5, E[Y] = 3, and V[Y] = 2. Define a new random variable T = 4X - 3Y. Compute the following. (a) E[T] (b) V[T] (c) P(T 25) (d) P(T > 42)