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A small farmer in a developing country grows cassava and groundnuts. The amount of cassava he grows per year is a random variable, X, with a mean of 500 pounds, and a variance of 625 (pounds-squared). The amount of groundnuts he grows per year is a random variable, Y, with a mean of 700 pounds, and a variance of 2500 (pounds-squared). The amounts of cassava and groundnuts he grows have a covariance of 475. The farmer uses 400 pounds of cassava and 300 pounds of groundnuts for consumption by his family. The remainder of each crop is sold for cash. He can sell cassava for \$2 per pound, and groundnuts for \$3 dollars per pound.

  1. What is the variance of the farmer’s cash income?

Really not sure where to go with this problem. I found the expected cash income but have no probability values to calculate the variance.

Xander Henderson
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Mary
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1 Answers1

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I saw you had a comment. I've edited to give more details:

$Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$

Farmers Income = $I=2X+3Y$

$Var(I)=Var(2X+3Y)=4Var(X)+9Var(Y)+2[(2)(3)Cov(X,Y)]=4(625)+9(2500)+12(475)=?$