Question 1

Which of the following best describes a continuous random variable?

Accepted Answer

A variable that can take any value within an interval, including all real numbers between two points.

Question 2

The probability that a continuous random variable $$X$$ takes any exact value (e.g., $$P(X = a)) is$$:

Accepted Answer

$$0$$

Question 3

Given a probability density function (PDF) $$f(x)$$ for a continuous random variable $$X$$, the probability that $$X$$ falls between $$a$$ and $$b is$$:

Accepted Answer

$$\int_a^b f(x) dx$$

Question 4

Which of the following is NOT a property of a probability density function (PDF) for a continuous random variable?

Accepted Answer

$$f(x)$$ can be negative for some $$x$$

Question 5

The expected value (mean) of a continuous random variable $$X$$ with PDF $$f(x) is $$given by:

Accepted Answer

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

Question 6

Suppose the height of adult males in a population is normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the $$z-$$score for a man who is 76 inches tall?

Accepted Answer

$$z = \frac{76 - 70}{3} = 2$$

Question 7

For a uniform distribution on the interval $$[a, b]$$, the probability density function is:

Accepted Answer

$$f(x) = \frac{1}{b-a}$$ for $$a \leq x \leq b$$

Question 8

A hospital randomly assigns patients to receive a drug at any time between 8:00 am and 3:00 pm. What is the probability that a patient receives the drug between 10:00 am and 12:00 pm?

Accepted Answer

$$\frac{2}{7}$$

Question 9

Which of the following statements about the normal distribution is FALSE?

Accepted Answer

It is defined only for positive values of $$x.$$

Question 10

If $$X \sim N(100, 16)$$, what is the probability that $$X is $$greater than 108? (Use the standard normal distribution.)

Accepted Answer

0.0228

Question 11

The cumulative distribution function (CDF) $$F(x)$$ for a continuous random variable $$X$$ gives:

Accepted Answer

The probability that $$X is $$less than or equal to $$x$$

Question 12

Suppose $$X is $$uniformly distributed on $$[2, 8]. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 5$$

Question 13

Which of the following real-world scenarios is best modeled by a uniform distribution?

Accepted Answer

The time a bus arrives at a stop, if it is equally likely to arrive at any time between 8:00 and 8:30 am.

Question 14

For a normal distribution $$N(\mu, \$$sigma^2$$)$$, the density function is:

Accepted Answer

$$f(x) = \frac{1}{\sqrt{2\pi\$$sigma^2$$}} e$$^{-\frac{(x-\mu)^2}$${2\$$sigma^2$$}}$$

Question 15

If $$X is a $$continuous random variable with PDF $$f(x)$$, which of the following statements is true?

Accepted Answer

$$P(a \u003c X \u003c b) = \int_a^b f(x) dx$$

Question 16

A laboratory measures the concentration of a drug in blood, which is normally distributed with mean $$50 mg/L $$and standard deviation $$10 mg/L. $$What is the probability that a randomly selected patient has a concentration above $$70 mg/L$$?

Accepted Answer

0.0228

Question 17

Which of the following best describes the relationship between the PDF and the CDF for a continuous random variable?

Accepted Answer

The PDF is the derivative of the CDF.

Question 18

Suppose $$X is $$uniformly distributed on $$[0, 10]. $$What is the probability that $$X is $$between $$3$$ and $$7$$?

Accepted Answer

$$0.4$$

Continuous Probability Distributions: Properties, Uniform and Normal Distributions

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