Applied Statistics for Business & Economics

Student ID: Name: STA 103 – Homework You will write your answers below (showing all necessary work), then upload a scan of your homework to Gradescope. 1. Let X and Y have the following joint distribution: X Y 0 1 0 2/12 1/12 1 1/12 1/12 2 5/12 2/12 (a) Find the probability distribution of X|Y = 0, and E[X|Y = 0]. (b) Find E[Y ] and E[X]. Final Numerical Answers Below (c) Find the probability that Y is larger than zero. Final Numerical Answer Below (d) Are X and Y dependent? Justify your answer. 1 2. Suppose we flip a fair coin four times. Let X = number of heads, Y = 1 if last flip is a tails. (a) Write down the joint distribution of X and Y . (b) Find E[X] and E[Y ]. Final Numerical Answers Below (c) Find E[XY ] Final Numerical Answer Below (d) Find the probability distribution for X|Y = 1. (e) Are X and Y independent? Justify your answer. 2 3. Let X and Y have the following joint distribution: X Y 0 2 4 -1 0.20 0.10 0.25 (a) Find the probability distributions for X and Y . (b) Find E[X] and E[Y ]. Final Numerical Answers Below (c) Find the probability that X is larger than 1. Final Numerical Answer Below (d) Find E[XY ] Final Numerical Answer Below 3 1 0.15 0.20 0.10 4. Let X and Y be dependent random variables, where E[X] = 4, σX = 3, E[Y ] = −2, σY = 6. Let ρ(X, Y ) = 0.50. Find the expected value and the variance of the following linear combinations. (a) W1 = 12X + 2Y + 2 Final Numerical Answers Below E[W1 ] V ar[W1 ] (b) W2 = (1/4)X − 20Y + 100 Final Numerical Answers Below E[W2 ] V ar[W2 ] (c) W3 = X − Y Final Numerical Answers Below E[W3 ] V ar[W3 ] (d) W4 = X + Y Final Numerical Answers Below E[W4 ] V ar[W4 ] 4 5. Let X and Y have the following joint distribution: X Y 0 2 4 Further, suppose σX = q 1664 625 , σY = q 0 5/50 10/50 10/50 1 8/50 1/50 10/50 2 1/50 5/50 0 3111 2500 . (a) Find Cov(X, Y ). Final Numerical Answer Below (b) Find ρ(X, Y ). Final Numerical Answer Below (c) Find Cov(1 − X, 10 + Y ) Final Numerical Answer Below (d) Find ρ(1 − X, 10 + Y ). Hint, you should use (c), and find V ar[1 − X], V ar[10 + Y ]. Final Numerical Answer Below 5