Question

A medium-sized Malaysian company needs to determine how large its cash reserve should be in order to assure its capability to service fixed charges. The probability distribution of earnings for the next quarter is estimated to be a normal distribution with mean RM 6 million and standard deviation RM 1.5 million. (Note: The unit of currency in Malaysia is the Ringgit, denoted by RM.) It has been determined that fixed charges in the next quarter will be RM 4 million.

(a) We shall denote the random variable "earnings for the next quarter" by X. Write down the formula for the standard normal variable Z corresponding to X.

(b) Using tables for the standard normal distribution, calculate the probability that earnings for the next quarter will exceed RM 8 million. Show your working.

(c) What is the probability that earnings for the next quarter will be within RM 1 million of the mean? Show your working.

(d) If the company maintains a cash reserve of RM 1 million, then what is the probability that the fixed charges will exceed the sum of its earnings and the cash reserve? Show your working.

Answer 1

Mean earnings, = RM 6 million

Standard deviation, = RM 1.5 million

Fixed charges = RM 4 million

(a)

->

(b) The required probability = P(X > 8)

= P{Z > (8 - 6)/1.5}

= P(Z > 1.33)

= 0.0918

(c) The required probability = P(|X - 6| ≤ 1)

= P(|Z| ≤ 1/1.5)

= P(|Z| ≤ 0.667)

= 0.495

(d) Probability (Fixed charges > Earnings + Cash reserve)

= P(4 > X + 1)

= P(X < 3) = P(Z < -2) = 0.0228