Question

Consider a company that has two different divisions. The annual profits from the two divisions are independent and have distributions Profit1 ~ N(5, 3^2) and Profit2 ~ N(7, 4^2) respectively. Both the profits are in $ Million. Assume that $1 = Rs. 45
A. Specify a Rupee range (centered on the mean) such that it contains 95% probability for the annual profit of the company.
B. Specify the 5th percentile of profit (in Rupees) for the company
C. Which of the two divisions has a larger probability of making a loss in a given year?

Answer 1

Answer:

a) Rs 603.68

b) Rs 476.33

c) first division

Step-by-step explanation:

Total profit = profit 1 + profit 2 = P

hence P ~ N(12,74) = $\frac{P-12}{\sqrt{74} } ~ N(0,1)$

A ) Specifying a Rupee range ( centered on the mean ) that contains 95% probability for annual profit of the company

$13.41 = Rs 603.68

B) specifying the 5th percentile of profit

p = $10.59 ≈ Rs 476.33

C) The  division that has a larger probability of making a loss in a given year is the first division

attached below is a detailed solution of the given problem

Answer 2

A): Range containing 95% probability for profit of company is

(Rs. 99M, Rs. 1026M).

B): Rs. 170.1 Million.

C): First division of the company has larger probability of making a loss.

Given that:

$1 = Rs. 45

$Profit_1 \sim N(5, 3^2) \\ Profit_2 \sim N(7, 4^2)$

Thus,

Company's profit:

$P \sim N( 5+7, 3^2 + 4^2) = N(12, 5^2)$

A):

95% of the probability lies between 1.96 standard deviations of the mean.

Thus range is:

$= (12 - 1.96\times 5, 12 + 1.96 \times 5)\\\\= (\ 2.2M, \ 22.8M)\\ \\= (\rm Rs. \: 99M, \rm Rs. \: 1026M)$

B): Fifth percentile is calculated as:

$P(Z \leq \dfrac{p-12}{5}) = 0.05$

From p values of z score table, we get:

$\dfrac{p-12}{5} = -1.644\\p = 12 - 8.22 = 3.78$

Thus at $3.78M dollars, or Rs. 170.1M amount, 5th percentile of profit lies.

Or 5th percentile of profit is Rs. 170.1 Million.

C): Loss is when profit < 0

Thus: p < 0

The first division of company, thus have larger probability of making a loss in a given year.

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