Question

). A farmer knows from experience that his wheat harvest  Q (in        

                     bushels) has the probability distribution given by:

                                         Q = 150  with probability  0.30 ,

                                              = 170    with probability  0.45,

                                               =200    with probability  0.25;

                      note that the probabilities add up to 1 as they are required by a probability   

                      distribution. Suppose the demand function he faces in the market place is given by:

                                              p  =  320  -  0.5  Q

                       where  p =  price in dollars per bushel. Let  R  = total revenue = p x Q. [Note: You may find it

                      convenient to first derive the probability distribution of  R.]

Find E ( R ).

                      Also, define his Profits as :

                                            Profits =  R - C  ,

                     where the total cost  C  (in dollars) is a function of  Q  given by

                                            C =  150  -   10 Q   + 2 Q².

Find  E(Profits).

Answer 1

Pdf of Q
O	P(Q)
150	0.3
170	0.45
200	0.25
total	1

  p  =  320  -  0.5  Q

R = p * Q  

= (320  -  0.5  Q) * Q

=

We know that R is dependent on Q so Rs probablties will also be dependent on Q's

For the distribution we will substitute each value of Q in R with the corresponding probability

Pdf of R
O	R	P(R)	R*P(R )
150	36750	0.3	11025
170	39950	0.45	17977.5
200	44000	0.25	11000
total		1	40002.5

Expected Value of R =

Exp (R) = 40002.5

Profits = R - C

C= 150  -   10 Q   + 2

We sub 'C' and 'R' formula in the profits

Profits =

=

Again now profits is a function of Q. So its probabilties will be dependent on Q.

Pdf of Profits
O	Profits	P(Profits)	Profits*P(Profits
150	80100	0.3	24030
170	95900	0.45	43155
200	121850	0.25	30462.5
total		1	97647.5

Expected profits =

E(profits) = 97647.5