Question 1

Which of the following represents a difference between continuous and discrete random variables?

Accepted Answer

All of the choices are true. Continuous random variables assume an uncountable number of values, and probability for them means finding the area under a curve. On the other hand, discrete random variables assume a countable number of values, and probability for them means summing individual probabilities. The probability for any individual value of a continuous random variable is zero, but for discrete random variables, it is not.

Question 2

A probability density function shows the probability for each value of X.

Accepted Answer

A probability density function shows the density of probabilities across a range of values, not the probability for each individual value of X. The probability of any given value is technically 0 since the probability is spread out across an infinite number of values. The area under the probability density function curve between two values represents the probability of X being within that range.

Question 3

To be a legitimate probability density function,all possible values of f(x)must lie between 0 and 1 (inclusive).

Accepted Answer

This is a fundamental property of any probability density function. It must be non-negative, meaning all possible values of f(x) must be greater than or equal to 0, and it must be normalized, meaning the area under the curve must be equal to 1. Therefore, all possible values of f(x) must lie between 0 and 1 (inclusive).

Question 4

In practice,we frequently use a continuous distribution to approximate a discrete one when the number of values the variable can assume is countable but large.

Accepted Answer

This statement is true. Using a continuous distribution to approximate a discrete one can often simplify calculations and make them more convenient.

Question 5

A continuous random variable X has a uniform distribution between 5 and 25 (inclusive),then P(X = 15)= 0.05.

Accepted Answer

A continuous uniform distribution assumes that any value between the minimum and maximum values has equal probability. Therefore, the probability of any single value (such as 15) in this range is always zero. Therefore, P(X = 15) cannot be 0.05 and the statement is false.

Question 6

If X is a continuous random variable on the interval [0,10],then P(X > 5)= P(X ≥ 5).

Accepted Answer

Since X is a continuous random variable, the probability that X takes any specific value is 0. Therefore, P(X>5) and P(X≥5) are equivalent and both equal to the probability that X falls within the interval (5,10].

Question 7

Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval.

Accepted Answer

Continuous probability distributions describe probabilities associated with random variables that can assume an infinite number of values within a given interval, not just a finite number.

Question 8

If X is a continuous random variable on the interval [0,10],then P(X = 5)= f(5)= 1/10.

Accepted Answer

Since X is a continuous random variable, the probability of X taking any particular value (including 5) is zero. Therefore, P(X = 5) = 0, and f(5) cannot equal 1/10.

Question 9

Since there is an infinite number of values a continuous random variable can assume,the probability of each individual value is virtually 0.

Accepted Answer

This is known as the "continuity property" of continuous random variables. Since there are infinitely many possible values, the probability of selecting any individual value is negligible. Instead, we focus on the probability of selecting a range of values, which can be expressed as the area under the probability density function.

Question 10

To be a legitimate probability density function,all possible values of f(x)must be non-negative.

Accepted Answer

This is one of the properties of a probability density function (PDF). It must always return non-negative values for all possible values of x.

Question 11

If a point y lies outside the range of the possible values of a random variable X,then f(y)must equal zero.

Accepted Answer

This statement is true by definition of probability density function. The pdf of a continuous random variable assigns a probability density to each possible value of the variable, such that the total area under the curve equals 1. Therefore, any point outside the possible range of values has zero probability density, and f(y) must equal zero.

Question 12

A continuous random variable X has a uniform distribution between 10 and 20 (inclusive),then the probability that X falls between 12 and 15 is 0.30.

Accepted Answer

This statement is true according to the properties of the uniform distribution. The probability of a continuous random variable falling within a given interval is proportional to the length of that interval, and since the possible values of X are uniformly distributed between 10 and 20, the length of the interval from 12 to 15 represents 3/10 of the total interval length, making the probability 0.3.

Question 13

A continuous random variable is one that can assume an uncountable number of values.

Accepted Answer

A continuous random variable can take on any value within a certain interval or range, and therefore has an uncountable number of possible values.

Question 14

A continuous probability distribution represents a random variable having an infinite number of outcomes which may assume any number of values within an interval.

Accepted Answer

A continuous probability distribution can take any value within a certain range, and this range can theoretically be infinite, resulting in an infinite number of possible outcomes.

Question 15

The probability density function,f(x),for any continuous random variable X,represents:

Accepted Answer

The probability density function, f(x), for a continuous random variable X represents the relative likelihood of different outcomes occurring. It does not represent the probability of X taking on a specific value, but rather the height of the density function at a given point (x) represents the probability density at that point. Therefore, choice C is the correct answer.

Question 16

The sum of all values of f(x)over the range of [a,b] must equal one.

Accepted Answer

The sum of all values of f(x) over the range of [a,b] does not necessarily equal one. It may equal any finite or infinite value or may not have a defined value at all.

Question 17

Let X represent weekly income expressed in dollars.Since there is no set upper limit,we cannot identify (and thus cannot count)all the possible values.Consequently,weekly income is regarded as a continuous random variable.

Accepted Answer

This statement correctly describes the characteristics of a continuous random variable.

Question 18

Which of the following is always true for all probability density functions of continuous random variables?

Accepted Answer

All the statements A, B, and C accurately describe characteristics of probability density functions for continuous random variables, making D the correct choice.

Question 19

We distinguish between discrete and continuous random variables by noting whether the number of possible values is countable or uncountable.

Accepted Answer

This statement is true. A random variable is said to be discrete if it can take on a countable number of values (such as the outcomes of rolling a fair six-sided die), while a continuous random variable can take on any value within a range (such as the amount of rainfall in a given area over a year).

Question 20

A continuous random variable X has a uniform distribution between 5 and 15 (inclusive),then the probability that X falls between 10 and 20 is 1.0.

Accepted Answer

The answer of A continuous random variable X has a...

Quiz 8: Continuous Probability Distributions