Q: For a continuous uniform distribution on $$[a, b]$$, what is the formula for the probability that $$X$$ falls between $$c$$ and $$d$$ where $$a \\leq c \u003c d \\leq b$$?

$$P(c \\leq X \\leq d) = \\frac{d-c}{b-a}$$

Question 1

Which of the following statements best describes the probability density function (PDF) for a continuous uniform distribution on the interval $$[a, b]$$?

Accepted Answer

The PDF is a constant value for all $$x in$$ $$[a, b]$$ and zero elsewhere.

Question 2

Given a continuous random variable $$X$$ uniformly distributed between 2 and 12, what is the value of its probability density function $$f(x)$$ for $$2 \leq x \leq 12$$?

Accepted Answer

$$f(x) = 0.1$$

Question 3

A call center agent's response time $$X is $$uniformly distributed between 2 and 12 seconds. What is the probability that a call is answered in less than 5 seconds?

Accepted Answer

$$P(X \u003c 5) = 0.3$$

Question 4

Which of the following is NOT a property of a valid probability density function (PDF)?

Accepted Answer

The function can take negative values for some $$x.$$

Question 5

A passenger train arrives at a station every 20 minutes. If a passenger arrives at a random time, what is the probability that the passenger will wait less than 10 minutes?

Accepted Answer

$$0.5$$

Question 6

For a continuous uniform distribution on $$[a, b]$$, what is the formula for the probability that $$X$$ falls between $$c$$ and $$d$$ where $$a \leq c \u003c d \leq b$$?

Accepted Answer

$$P(c \leq X \leq d) = \frac{d-c}{b-a}$$

Question 7

Which of the following curves represents a valid probability density function for a continuous random variable?

Accepted Answer

A curve with total area under it equal to 1 and all values non-negative.

Question 8

If $$X is a $$continuous random variable with a uniform distribution on $$[0, 8]$$, what is $$P(2 \leq X \leq 4)$$?

Accepted Answer

$$0.25$$

Question 9

Which of the following best explains why the probability of a continuous random variable taking any exact value is zero?

Accepted Answer

Because the area under the PDF at a single point is zero.

Question 10

A continuous random variable $$X$$ has a uniform distribution on $$[3, 9]. $$What is the expected value $$E[X]$$?

Accepted Answer

$$6$$

Question 11

Suppose a continuous random variable $$X$$ has a uniform distribution on $$[a, b]. $$What is the variance $$Var(X)$$?

Accepted Answer

$$\frac{(b-a)^2$}{12}$$

Question 12

A local call center tracks the response time for customer service agents to answer incoming calls during peak hours. If the response time is uniformly distributed between 2 and 12 seconds, what is the probability that a call is answered between 6 and 10 seconds?

Accepted Answer

$$0.4$$

Question 13

Which of the following is true about the shape of the probability density function for a continuous uniform distribution?

Accepted Answer

It is a rectangle with constant height over the interval $$[a, b].$$

Question 14

If the total area under a curve representing a probability density function is less than 1, what does this indicate?

Accepted Answer

The function is not a valid probability density function.

Question 15

A continuous random variable $$X is $$uniformly distributed on $$[2, 12]. $$What is the probability that $$X is $$greater than 8?

Accepted Answer

$$0.4$$

Question 16

Which of the following best describes the difference between discrete and continuous random variables?

Accepted Answer

Discrete random variables take on countable values, while continuous random variables take on any value in an interval.

Question 17

If $$X is a $$continuous random variable with a uniform distribution on $$[a, b]$$, what is the height of the probability density function?

Accepted Answer

$$\frac{1}{b-a}$$

Question 18

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

Accepted Answer

$$0.4$$

Question 19

Which of the following is a real-world example of a continuous uniform distribution?

Accepted Answer

The time a bus arrives at a stop, uniformly distributed between 8:00 and 8:30 AM.

Continuous Random Variables and Uniform Distributions

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