ECEN 3810 Midterm #2 Professor David G. Meyer Date: November 15, 2010 Abstract Instructions: This is a closed-book exam. No books, class notes, or HWs are allowed. You can use a simple calculator (arithmetic only) and your one 8.5 × 11 sheet of notes. Show work and explain your steps to receive full credit. “Magically” appearing answers with no work or explanation aren’t acceptable. Start each problem on a new page. Use only one side of the paper. Work & write neatly! Warning! Failure to heed these style requirements can result in points being deducted. Good neatness and style can result in extra points being awarded. Sign the honor code pledge below and turn this exam page in with your work, or write the pledge and print your name on your own paper and sign it there. Either way, don’t forget the pledge as the work cannot be graded without it. On my Honor as a University of Colorado at Boulder Student, I have Neither Given nor Received Unauthorized Aid on This Examination Signed: PRINT NAME: 1 1. (30 pts) We independently and randomly choose two numbers, X1 and X2 , greater or equal to zero and less than or equal to two. That is, X1 and X2 are continuous, independent random variables, both uniformly distributed on [0, 2]. (a) (5 pts) The variance of X1 is 1/3. What is E [X1 ]? (b) (5 pts) What is Cov [X1 , X2]? Why? (c) (10 pts) Find and write the joint PDF, fX1 ,X2 (x1, x2). (d) (10 pts) What is the probability that the sum of the squares of X1 and X2 is less than 1? In other words, find P X12 + X22 < 1 2. (45 pts) We perform the following experiment: We flip a fair coin. If it comes up “heads” we then randomly draw one ball from an urn with 1 red and 1 white. If the coin comes up “tails” we randomly draw one ball from an urn with 3 white and 1 red. We define two random variables, X1 , and X2 , on this experiment as follows. 1 If coin flip is heads 1 If red ball drawn X1 = X2 = (so X2 is just the # of reds drawn) 0 If coin flip is tails 0 If white ball drawn (a) (6 pts) X1 and X2 are clearly NOT independent. Would you expect them to be negatively correlated, positively correlated, or uncorrelated? Explain. (b) (8 pts) The joint PDF, fX1 ,X2 (x1 , x2), is fX1 ,X2 (x1 , x2) = 3 1 δ(x1)δ(x2 ) + a10δ(x1 − 1)δ(x2) + δ(x1 )δ(x2 − 1) + a11δ(x1 − 1)δ(x2 − 1) 8 8 What is the numerical value for the coefficient a11? Show all your work. (c) (8 pts) Find and write the marginal PDF, fX2 (x2). (Note: You may leave “a10” as an undetermined coefficient in your answer). (d) (8 pts) Clearly E [X1 ] = 1/2. What is E [X2 ]? (e) (15 pts) Compute Cov [X1 , X2]. 3. (25 pts) We have an experiment with sample space S and an event, E ⊆ S, that we are interested in. We can independently repeat the experiment multiple times and we’d like to figure out P [E]. We hit upon the idea of defining an “indicator” RV X as follows: 1 If event E occurs X= 0 If event E c – complement of E – occurs (hence event E did not occur) The PDF of this “indicator” RV X is then fX (x) = P [E] δ(x − 1) + P [E c ] δ(x) = pδ(x − 1) + (1 − p)δ(x) where p = P [E] is the unknown parameter we want to figure out. (a) (5 pts) Show carefully and completely that E [X] = p. (b) (5 pts) Compute Var [X]. Show your complete work and derivation(s). (c) (15 pts) Worst case, how many repetitions, N , of the experiment should we do, and repetitive measurements of X should we take, to estimate p = E [X] to within ±.01 to 95% confidence using the sample mean, MN (X)? (d) (3pts bonus) What is true about the events E and E c when the worst case occurs? 2

Download PDF

advertisement