3. Bivariate distributions This problem follows the example on minute 24:00 of the Ch5 part 1 video. You invest equal sums (let's say $1 and $1) into "blue chip" shares and penny stock. And worry about the volatility of your portfolio. The table below lists the probabilities of getting certain values of returns: Return on Probability Return on blue Probability penny stocks, $ chips, $ -10 0.3 10 0.1 10 0.2 10 0.7 30 0.5 30 0.2 For penny stocks, the mean return is up = $14 and the variance is as = $ 304. For blue chips, they are us = $12 and = = $ 116. The joint probability table for them is as follows: Return on blue chips -10 10 30 Return 10 0.04 0.2 0.06 on penny 10 0.02 0.14 0.04 stocks 30 0.04 0 .36 0.1 (a) Compose the bivariate table that has the possible values for P+B in the first column and their respective probabilities in the second column. Double-check that the probabilities add up to one. (b) Find the mean and variance of your total portfolio return, Ap+s and of+s. (c) Use the "square of the sum" formula for bivariate variance to find the covariance, ops. (d) Now suppose you want to reduce the volatility of your portfolio, even if at the cost of the expected return. You want to only have a = 0.67 in your penny stocks and as much as b = 1.33 in your blue chips. Use the linear combination formula to find the variance of your new portfolio, Jap+bs