Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of $1\mathrm{.}4$ on the market index. Firm$-$specific returns all have a standard deviation of $34\%.$

Suppose that an analyst studies $20$ stocks and finds that one$-$half of them have an alpha of $+2\mathrm{.}2\%,$ and the other half have an alpha of $2\mathrm{.}2\%.$ Suppose the analyst invests $$1\mathrm{.}0$ million in an equally weighted portfolio of the positive alpha stocks, and shorts $$1$ million of an equally weighted portfolio of the negative alpha stocks.

a$.$ What is the expected profit $($in dollars$)$ and standard deviation of the analyst$$s profit? $($Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.$)$

b$.$How does your answer change if the analyst examines $50$ stocks instead of $20$ stocks? $100$ stocks? $($Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.$)$