Mathematical statistics homework

EC 233 Final Exam, Fall 2020

Ozan Hatipoğlu

Answer question 1 and choose one of second or third question. ( 2 total). Each question has the same points (50) but a different degree of difficulty and a time penalty. The earlier you post the higher the grade you will get out of its potential. If you submit as late as still possible( example: submitting 1 second earlier than the deadline) at most x points will be subtracted out of total. The penalty , x, is linear in time passed. You can not submit twice. Make sure everything is OK before you click the submit button. No questions about any of the details of this exam will be answered. If you think a question is wrong, misleading or missing necessary info, just state it in your answer. Upload 1 pdf file with handwritten answers. You can use R , Excel or other software. By submitting you agree that you accept the honor code in Midterm. Deadline: February 12, 23.59:00 PM.

1) (x= 5) You are a portfolio manager. You observe that two normally distributed asset returns, " and " have the same mean, " , and the same variance, ! , but are independent. Given that " find the mean and riskiness (variance) of the return of the portfolio, R, that consists " of Asset 1 where " and " of Asset 2 where " , i.e. (R =a" +b" ) and " . First determine " randomly by selecting from a uniform distribution between (0,1), (" ), then solve the question. Write all the details of the steps that you actually went through while solving this exercise in a very clear manner.

2) (x= 15) , Choose two numbers, " and " randomly between 0 and 1000 and determine c such that " serves as a continuous probability density function for " where " . Using " , calculate " and " .

3) (x=5) Given the joint distribution

" , find a possible value for " first. Then determine one number b randomly in any meaningful way you like. (" and calculate

i)"

ii )" - "

Make sure to write down all the details that you actually went through while solving this exercise in a very clear manner.

R1 R2 μ σ P (R1 < 0.05) = 0.2,

a 0 < a < 1 b 0 < b < 1 R1 R2 a + b = 1 a

b = 1 − a

n1 n2 f (x) = c x x n1 < x < n2

f (x) = c x E (2x + 1) v a r (2x + 1)

f (x1, x2) = a x1x2 f or 0 < x1 < 1, 0 < x2 < 1, a n d 0 el s e wh er e a b ≠ 0)

P (b X1 + b X2 ≤ 1)

E (X1 ∣ X2 = 1) (E (X1 ∣ X2 = 0.5))