Question

David’s firm is planning a major investment. The amount of gross profit, indicated by RV 'X' (in millions of dollars) is uncertain, but an estimate gives the following empirical probability distribution: Profit 1 1.5 2 4 10 Probability 0.1 0.2 0.4 0.2 0.1 David’s firm owes its venture capital lender a fee of 0.2 million plus 10% of the gross profit X. The retained profit is for David's firm is the RV 'Y'.i. Calculate the mean and standard deviation of the gross profit. ii. Calculate the coefficient of variation of the gross profit. iii. Calculate the mean and standard deviation of the retained profit. iv. Calculate the coefficient of variation of the retained profit.

Answer 1

Solution :

Given probabiliy distribution for random variable x is,

Profit x	1	1.5	2	4	10
probability p(x)	0.1	0.2	0.4	0.2	0.1

iMean and sd for gross profit

.

= (1*0.1)+(1.5*0.2)+(2*0.4)+(4*0.2)+(10*0.1)= 3

standard deviation

= sqrt(0.4+0.45+0.4+0.2+4.9)

= 2.52

ii. coefficient of variation = (sd/mean)*100 = (2.52/3)*100 = 84%

iii.Mean and sd for retailed profit.

here given that ,David’s firm owes its venture capital lender a fee of 0.2 million plus 10% of the gross profit X.

So Y = 0.9X - 0.2

So using formula as, for y =ax+b, = a + b and = a*

Mean = = 0.9 - 0.2 = (0.9*3)-0.2 = 2.5

Standard deviation = = 0.9 = 0.9 *2.52 = 2.268

iv) coefficient of variation = (sd/mean)*100 = (2.268/2.5)*100 = 90.72%