3 Farmer Njoroge\'s Market Njoroge the farmer has a Ibs of apples and b Ibs of p

Question

3 Farmer Njoroge's Market Njoroge the farmer has a Ibs of apples and b Ibs of potatoes for sale. The market price, in Kisumu Kenya, for apples each day is a randonn variable with a mean of , dollars and a standard deviation of Oz dollars. Similarly, for a pound of potatoes, the mean price is , dollars and the standard deviation is ay dollars. Assume that the market prices for potatoes and apples has a correlation of p. It costs Njoroge d dollars to bring all the apples and potatoes to market. Assume Njoroge will sell all of each type of produce each day ise t a, b, d and the parameters HmPmOz,Oy, p. b) Find the mean and the variance of the net income, also in terms of the given constants parameters. and c) Njoroge is a popular name in Kenya. Say, for any Njoroge selling potatoes and apples at the market the correlation, is distributed uniform between .5 and .75 Generate 100 random Nyroges-here pis a randon quantity-with a 5, b = 10 and 100 more random Njoroges with a- -5,b 10. Set the means and variances to 0 and 1, say. Generate appropriate plots. Talk about what you see. Hint: set Pv, etc. equal to numbers to see how it works out. Then erpress your final ansver, generally, in symbols 3 5 6 7 8 9 R T Y U F G H J K L

Explanation / Answer

(a) Random variable: A random variable is a real valued function defined on the sample space such that it associates a real number to each elementin the sample space.

Let X be a random variable which denotes the market price of apples on any particular day, it is given that the mean price of apples on any day is µX and the standard deviation of the price of apples on any day is X.

Let Y be a random variable which denotes the market price of potatoes on any particular day, it is given that the mean price of potatoes on any day is µY and the standard deviation of the price of potatoes on any day is Y.

Njoroge's net income is given by N= aX + bY - d

(b) The mean and variance of the net income is given by E(N)=E(aX+bY-d)

E(N) = aE(X) + bE(Y) -d

E(N) = aµX + bµY-d

and the variance is given by:

V(N) = V(aX+bY-d) =a2V(X) + b2V(Y) + 2abCov(X,Y)

V(N) = a2(X)2 + b2(Y)2 + 2abXY

(c) The plots can be created by feeding the values in the above equations.

For the first 100 simulations, a=5 and b=10, and mean and variance of each random variable is 0 and 1 respectively.

For the next 100 simulations, a=-5 and b=10, and mean and variance of each variable is 0 and 1 respectively.

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