Jindal School of Management The University of Texas at Dallas

Analytics of Finance

Assignment 1 - due 09/06/2021 (Tuesday) at 4PM

For additional help on R, go to my elearning site. For questions that involve R coding, you need to upload your codes in addition to your solutions that report your results and discussion in order to receive full credit. Please do not jhand in your computer printouts. Your solution file should be in PDF format according to naming specification in syllabus.

1. (Linear Algebra Review)
   1. Given two matrices: A = [ 3 2 1 1 2 1 ] and B = 3 2 1 2 2 1 , i) Compute C = 2A B and D = C B by hand; ii) Find |D| and D1 by hand.
   2. In R, set the seed to your student ID number using the command "set.seed(YourStudentID)", you can then generate 24 random numbers according to uniform distribution use command "runif(24)". Using the first 12 numbers to form a (3 by 4) matrix G1 and the rest of numbers to form a (4 by 3) matrix G2. Using R to i) compute H1 = G1G2 and H2 = G2G1; Comparing H1 and H2, what can you can you say about communicative product of a product? ii) Find |H1| and |H2| using the "det()" command; iii) Compute H1 1 using the "solve()" command. Can you compute H1 2 , and why?
2. (Conditional Probability) The first step to combat the Covid-19 is to develop an effective test kit. Suppose on average there are 1% chance being infected by the virus. One of the test being used will show negative for 99.5%
   • f uninfected people. In addition, the test has only 0.4% of "false negative" rate.
   • If you are tested positive, what is the chance that you are actually infected by the virus? What does your answer mean?
   • If you are tested negative, what is the chance that you are actually infected by the virus? What does your answer mean?
3. (Statistics Review)
   1. Suppose a random variable x follow a normal distribution of N [0, (13)2]. Let z = 9x2, what is the distribution of z? What is its mean, i.e. E(z)? [Hint: a2x2 = (ax)2 and use the property of the normal distribution and Chi-square distribution.]
   2. Suppose Xn = {x1, x2, , xn} is a sample with n observations drawn from the distribution of x in (a). For each observation xi, you can compute the corresponding zi according to the relation in (a). With {z1, z2, , zn}, you can then compute the sample average zn = (1/n)n i=1 zi as the mean estimate of z. For a sample size of n = 5, draw a radom sample Xn and compute an absolute error of the mean estimate for z according to, e(1)5 = |z5(1) E(z)|. Discard your sample
      • f Xn and redraw a new 5-observation sample for Xn. Compute and denote your second round

error estimate as e(2)5 . Repeat the exercise for 100 times. You need to report your average error, 5 = (e(1)5 + e 5 (2) + + e 5 (100) )/100

(c) Repeat (b) for n = 10, 100, and n = 500, respectively. With respect to the Law of Large Numbers, what can you say?