Question 1

Which of the following is NOT a requirement for a function $$f(x) to be a $$probability density function (pdf)?

Accepted Answer

$$f(x)$$ must be continuous for all real $$x$$

Question 2

Suppose the length of telephone calls (in minutes) is a continuous random variable with pdf $$f(x) on$$ $$[0, \infty). If$$ $$\$$int_2$$^3 f(x)dx = 0.18$$, what does this value represent?

Accepted Answer

The probability that a call lasts between 2 and 3 minutes

Question 3

Given the normal probability density function $$f(x) = \frac{1}{2\sqrt{2\pi}} e$$^{-\frac{(x-9)^2}$${8}}$$, what is the probability that $$x is $$between 3 and 7? (Refer to the calculator image provided.)

Accepted Answer

0.16

Question 4

A trust pays you $2,000 per year for 10 years as a continuous income stream. What is the total income you will receive by the end of 10 years?

Accepted Answer

$$\$$int_0$$^{10} 2000\,dt$$

Question 5

If the rate of income from a vending machine is $$f(t) = 5000e$$^{0.04t}$$, what is the total income produced during the first 5 years?

Accepted Answer

$$\$$int_0$$^5 5000e$$^{0.04t}$$\,dt$$

Question 6

The future value of a continuous income stream $$f(t)$$, continuously invested at rate $$r$$ for $$T$$ years, is given by which formula?

Accepted Answer

$$FV = \$$int_0$$^T f(t)e$$^{r(T-t)}$$\,dt$$

Question 7

If you receive $2,000 per year for 10 years, continuously invested at 8% compounded continuously, what is the future value at the end of 10 years?

Accepted Answer

$$\$$int_0$$^{10} 2000e$$^{0.08(10-t)}$$\,dt$$

Question 8

Given the price-demand equation $$p = D(x) = 20 - 0.05x$$, what does the consumers’ surplus at price $$p = 8$$ represent?

Accepted Answer

The total amount consumers are willing to pay above $$8$$

Question 9

For the supply function $$p = S(x) = 2 + 0.0002x^2$, what is the producers’ surplus at price p = 20$?

Accepted Answer

The area between $$p = 2 + 0.0002x^2$ and p = 20$ from x = 0$ to the quantity supplied at p = 20$

Question 10

Given $$p = D(x) = 20 - 0.05x$$ and $$p = S(x) = 2 + 0.0002x^2$, what is the equilibrium quantity?

Accepted Answer

$$x = 200$$

Question 11

If the total income from a continuous income stream over $$[a, b] is $$represented by the area under $$y = f(t)$$, which integral gives this total income?

Accepted Answer

$$\int_a^b f(t)\,dt$$

Question 12

A manufacturer finds that the life expectancy of its LED lights is normally distributed with mean 9 years and standard deviation 2 years. What is the probability that a randomly selected light lasts between 3 and 7 years?

Accepted Answer

0.16

Question 13

If the future value of a continuous income stream is $$30,639$$ and the total income received is $$20,000$$, how much interest was earned?

Accepted Answer

$$10,639$$

Question 14

Which of the following best describes the equilibrium price in a free competitive market?

Accepted Answer

The price at which supply equals demand

Question 15

Given the supply and demand equations $$p = D(x) = 20 - 0.05x$$ and $$p = S(x) = 2 + 0.0002x^2$, what is the equilibrium price?

Accepted Answer

$$p = 10$$

Question 16

If the rate of flow of a continuous income stream is $$f(t) = 5000e$$^{0.04t}$$, and the income is continuously invested at 12% for 5 years, what is the future value?

Accepted Answer

$$\$$int_0$$^5 5000e$$^{0.04t}$$e^{0.12(5-t)}$$\,dt$

Question 17

Which of the following integrals represents the consumers’ surplus at price $$p_0$$ for a price-demand function $$p = D(x)$$?

Accepted Answer

$$\$$int_0$$^{x_0$} [D(x) - p_0$]dx$$

Question 18

Which of the following is a correct interpretation of the area under a probability density function $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

The probability that the random variable $$x$$ falls between $$a$$ and $$b$$

Applications of Integration in Business and Economics

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