The amount of money that Jon can save after working for a summer is a random variable S with a mean of $1,500 and a standard deviation of $100.After saving this money Jon plans to go on a trip to India.He will change his money into Rupees at an exchange rate of 43 Rupees to one Dollar.This money he will bring to India.When he arrives in India he will buy a used motorbike.The price in India of a motorbike of the type he wants is a random variable B with a mean of 17,500 Rupees and a standard deviation of 600 Rupees.The amount of money Jon will have left (in Rupees)after changing his savings into Rupees and buying a motorbike in India is a random variable P which can be expressed in terms of S and B as P = 43S - B.Find the mean and standard deviation of P in Rupees.Assume that Jon's savings and the price of the bike are independent.
A)μ = 82,000 Rupees,σ = 4,900 Rupees
B)μ = 47,000 Rupees,σ = 3,700 Rupees
C)μ = 47,000 Rupees,σ = 4,342 Rupees
D)μ = 47,000 Rupees,σ = 4,258 Rupees
E)μ = 82,000 Rupees,σ = 889 Rupees

Question

Quizplus · Accepted Answer

Using the properties of expected value and standard deviation, we can find the mean and standard deviation of P:

E(P) = E(43S - B) = 43E(S) - E(B) = 43*1500 - 17500 = 47000

Var(P) = Var(43S - B) = 43^2*Var(S) + Var(B) = 184900 + 360000 = 544900

SD(P) = sqrt(Var(P)) = sqrt(544900) = 737.87 ≈ 734.2

Thus, the mean and standard deviation of P in Rupees are μ = 47,000 and σ = 4,342 respectively, which is choice C.

The Amount of Money That Jon Can Save After Working