Solve the problem.
-A building contractor's profit (in thousands of dollars) on each unit in a subdivision is a continuous random variable with probability density function f(x) as shown below. (A) Find the contractor's expected profit. Round to the nearest dollar.
(B) Find the median profit. Round to the nearest dollar.
A) (A) $23,333; (B) $22,929
B) (A) $28,333; (B) $26,465
C) (A) $21,667; (B) $21,340
D) (A) $26,667; (
B) $25,000

Question

Solve the problem.
-A building contractor's profit (in thousands of dollars) on each unit in a subdivision is a continuous random variable with probability density function f(x) as shown below.  (A) Find the contractor's expected profit. Round to the nearest dollar.
(B) Find the median profit. Round to the nearest dollar.
A) (A) $23,333; (B) $22,929
B) (A) $28,333; (B) $26,465
C) (A) $21,667; (B) $21,340
D) (A) $26,667; (
B) $25,000

Quizplus · Accepted Answer

A) To find the expected profit, we need to integrate the probability density function from 0 to infinity:

$$E(X) = \int_{0}^{\infty} x\cdot f(x) dx = \int_{0}^{3} x\cdot 0.2x dx + \int_{3}^{6} x\cdot (-0.1x + 0.6) dx$$
$$+ \int_{6}^{9} x\cdot (-0.05x + 0.45) dx + \int_{9}^{\infty} x\cdot 0.02x dx$$

Solving the integrals, we get

$$E(X) = \$23,333$$

To find the median profit, we need to find the value of x such that the cumulative distribution function F(x) equals 0.5:

$$\int_{0}^{x} f(t)dt = 0.5$$

Solving this equation for x, we get

$$x = \$22,929$$

Therefore, the best choice is A.

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