Question 1

Which of the following statements about a continuous random variable is TRUE?

Accepted Answer

Its probability is defined over intervals, not at individual points.

Question 2

Let $$f(x) be a $$probability density function (PDF) for a continuous random variable $$X. $$Which property must $$f(x)$$ satisfy?

Accepted Answer

$$\int_{-\infty}^{\infty} f(x) dx = 1$$

Question 3

Suppose the shelf life (in days) of a drug is modeled by the PDF $$f(x) = \frac{200}{10000 - x^2$}$$ for $$0 \leq x \u003c 100$$, and $$f(x) = 0$$ otherwise. What is the probability that the drug lasts at least 20 days?

Accepted Answer

$$\int_{20}^{100} f(x) dx$$

Question 4

Given a continuous random variable $$X$$ with PDF $$f(x)$$, the cumulative distribution function (CDF) $$F(x) is $$defined as:

Accepted Answer

$$F(x) = \int_{-\infty}^{x} f(t) dt$$

Question 5

Which of the following is NOT a property of a cumulative distribution function (CDF) $$F(x)$$ for a continuous random variable?

Accepted Answer

$$F(x)$$ can decrease for some $$x$$

Question 6

A city’s daily electricity consumption (in millions of kilowatt-hours) is modeled by the PDF $$f(x) = 0.02 - 0.002x$$ for $$0 \leq x \leq 10. $$What is the expected daily consumption?

Accepted Answer

$$\int_{0}^{10} x f(x) dx$$

Question 7

For a continuous random variable $$X$$ with PDF $$f(x)$$, which formula gives the variance $$V(X)$$?

Accepted Answer

$$V(X) = \int_{-\infty}^{\infty} (x - \mu)^2$ f(x) dx$$

Question 8

A probability density function $$f(x) is $$defined as $$f(x) = kx$$ for $$0 \leq x \leq 2$$, and $$f(x) = 0$$ otherwise. What is the value of $$k$$?

Accepted Answer

$$k = \frac{1}{2}$$

Question 9

The median $$m of a $$continuous random variable $$X$$ with PDF $$f(x) is $$defined as the value such that:

Accepted Answer

$$\int_{-\infty}^{m} f(x) dx = \frac{1}{2}$$

Question 10

Suppose $$f(x) is a $$PDF for $$X$$ and $$F(x) is $$its CDF. What is $$P(a \u003c X \u003c b)$$?

Accepted Answer

$$F(b) - F(a)$$

Question 11

If $$f(x) is a $$PDF defined as $$f(x) = \frac{1}{2}$$ for $$1 \leq x \leq 3$$, and $$f(x) = 0$$ otherwise, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \int_{1}^{3} x \cdot \frac{1}{2} dx$$

Question 12

Which of the following is a valid interpretation of the area under a probability density function $$f(x)$$ between $$a$$ and $$b$$?

Accepted Answer

It represents the probability that $$X$$ falls within the interval $$[a, b].$$

Question 13

A continuous random variable $$X$$ has PDF $$f(x) = \frac{1}{4}$$ for $$0 \leq x \leq 4. $$What is the standard deviation $$\sigma of$$ $$X$$?

Accepted Answer

$$\sigma = \sqrt{\int_{0}^{4} (x - 2)^2$ \cdot \frac{1}{4} dx}$$

Question 14

For a continuous random variable $$X$$ with PDF $$f(x)$$, which of the following is TRUE?

Accepted Answer

$$P(X = a) = 0$$ for any real number $$a$$

Question 15

A pharmaceutical company wants to know the probability that a drug with shelf life modeled by $$f(x) = \frac{200}{10000 - x^2$}$$ for $$0 \leq x \u003c 100$$ lasts more than 80 days. Which integral should they compute?

Accepted Answer

$$\int_{80}^{100} f(x) dx$$

Question 16

If $$f(x) is a $$PDF and $$F(x) is $$its CDF, which of the following is always TRUE?

Accepted Answer

$$F(x) is $$always increasing or constant.

Question 17

The variance of a continuous random variable $$X$$ can be computed using the alternative formula:

Accepted Answer

$$V(X) = E[X^2$] - (E[X])^2$

Question 18

A continuous random variable $$X$$ has PDF $$f(x) = kx$$ for $$0 \leq x \leq 2. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \int_{0}^{2} x \cdot kx dx$$

Continuous Random Variables, Probability Density Functions, and Statistical Measures in Business Calculus

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