Question

You are a small farmer in a poor Latin American country. You are trying to decide whether to continue to cultivate a traditional strain of wheat or switch to a new high-yield variety. If you plant the traditional strain, the yield of your wheat field (X) is approximately normally distributed with mean 60 bushels and standard deviation 6 bushels. If you plant the high-yield variety, the yield of your wheat field (Y) is approximately normally distributed with mean 70 bushels and standard deviation 15 bushels. Your family requires 50 bushels of wheat. Let’s assess the relative riskiness of the traditional strain and the high-yield variety.

a. If you grow the traditional variety of wheat, the probability that the yield will be less than 50 bushels is . FOUR DECIMALS.

b. If you grow the high-yield variety of wheat, the probability that the yield will be less than 50 bushels is . FOUR DECIMALS.

In addition to wheat, you also grow bananas. Your banana yield (W) is approximately normally distributed with mean 100 bushels and standard deviation 4 bushels. If you grow the traditional variety of wheat, your wheat yield and banana yield have a correlation of 0.5. However, if you grow the high-yield variety, your wheat yield and banana yield have a correlation of – 0.5 (a negative correlation). You sell all bananas (your family does not need any bananas), plus any wheat in excess of the 50 bushels for your family. The market price of wheat is $1 per bushel and the market price of bananas is $3 per bushel (in other words, you get $1 per bushel for the wheat and $3 per bushel for the bananas you sell). You need to earn $300 to pay the landlord rent on your farm. Let’s compare the probabilities that you’ll earn at least enough income to pay your rent.

c. The covariance between X and W is . The covariance between Y and W is . INTEGERS (NO DECIMALS).

d. Use R to represent income from selling wheat and bananas. R as a function of X and W is R= . R as a function of Y and W is R= . EQUATIONS, NO SPACES.

e. If you grow the traditional variety of wheat, your expected income (R) is and the variance of income is . INTEGERS (NO DECIMALS).

f. If you grow the high-yield variety of wheat, your expected income (R) is and the variance of income is . INTEGERS (NO DECIMALS).

g. If you grow the traditional variety of wheat, the probability that your income will be at least $300 is . FOUR DECIMALS.

h. If you grow the high-yield variety of wheat, the probability that your income will be at least $300 is . FOUR DECIMALS.

Answer 1