Suppose you are going to invest $$1000$ in some stocks and hold

them for one year. Suppose $$Safe Corp.$$ stock currently costs $$100.$

Its price a year from now is a random variable X with mean $$105$

and standard deviation of $10.$Risky Business$$ stock also currently

costs $$100.$ But its price a year from now is a random variable Y

with mean $$110$ and standard deviation of $20.$ Suppose the correla$-$

tion coefficient of X and Y is $0\mathrm{.}2.$

$($a$)$ How much should you invest in each stock if you want to max$-$

imize the expected value of what your stocks are worth one year

from now?

$($b$)$ How much should you invest in each stock if you want to mini$-$

mize the variance of what your stocks are worth one year from now?

NOTE: You are allowed to buy part of a stock. So for example,

you could invest $$333\mathrm{.}33$ in $$Safe Corp.$$ and $$666\mathrm{.}67$ in $$Risky

Business.$($It may be useful to use the formula:

var$($X$+$Y$)=$ var$($X$)+$ var$($Y$)+2$cov$($X$,$Y$)$