Final answer:
The distribution of the total stock value when buying one share of each of the four independent identically distributed normal stocks is N(80, 400). The probability of the total stock value exceeding $100 can be found using the CDF of the normal distribution. If one buys two shares of stock 1 and two shares of stock 2, the resulting distribution is still N(80, 400).
Step-by-step explanation:
Distribution of the Total Stock Value
When investing in four stocks where each stock, Xi, follows a normal distribution N(20, 100), we need to find the total distribution for the value of your stocks one year from now. Since these stocks are independent random variables, the sum of these normally distributed variables is also normally distributed. Specifically, for (a), the sum of four independent identically distributed (iid) normal random variables will have the distribution N(4*20, 4*100) due to the properties of the normal distribution, which is N(80, 400).
Probability of Total Stock Value Exceeding $100
For (c), the probability can be found using the cumulative distribution function (CDF) of the normal distribution or by using standard statistical software. For (e), after purchasing four shares of stock 1, the distribution of this investment is N(4*20, 4*100) which is N(80, 400) as well. The probability calculation will be similar to part (c).
Distribution After Purchasing Different Shares
For (f), buying two shares of stock 1 and two shares of stock 2 (each with the distribution N(20, 100)), the distribution for the value of the stocks is N(2*20 + 2*20, 2*100 + 2*100) because of the independence and identical distribution of the stocks. Therefore, the distribution is N(80, 400).