Question

building basic skills and vocabulary

1. what is a random variable? give an example of a discrete random variable

and a continuous random variable. justify your answer.

1. is the expected value of the probability distribution of a random variable

always one of the possible values of ( x )? explain.

true or false? in exercises 5–8, determine whether the statement is true or
false. if it is false, rewrite it as a true statement.

1. in most applications, continuous random variables represent counted data,

while discrete random variables represent measured data.

1. the mean of the random variable of a probability distribution describes how

the outcomes vary.

graphical analysis in exercises 9–12, determine whether the graph on
the number line represents a discrete random variable or a continuous random
variable. explain your reasoning.

1. the attendance at concerts for

a rock group

1. the distance a baseball travels

after being hit

using and interpreting concepts
discrete variables and continuous variables in exercises 13–18,
determine whether the random variable ( x ) is discrete or continuous. explain.

1. let ( x ) represent the number of cars in a university parking lot.

1. let ( x ) represent the volume of blood drawn for a blood test.

1. let ( x ) represent the number of texts a student sends in one day.

18.

Explanation:

1. A random variable assigns numbers to experiment outcomes. Discrete variables have countable values; continuous variables have infinite, measurable values.
2. The expected value is a weighted average, not always a possible outcome.
3. Discrete variables represent counted data; continuous represent measured data.
4. The mean measures central tendency, not outcome variation.
5. Concert attendance is countable, so discrete.
6. Baseball travel distance is measurable, so continuous.
7. Number of cars is countable, so discrete.
8. Blood volume is measurable, so continuous.
9. Number of texts is countable, so discrete.

Answer:

1. Random variable definition: A variable whose value is determined by the outcome of a random experiment.

• Discrete example: Number of heads in 3 coin flips (countable values: 0,1,2,3).
• Continuous example: Height of a randomly selected person (can take any value in a range, e.g., 150.2 cm, 175.5 cm).

1. No, the expected value is not always a possible value of $x$.

• Explanation: It is a weighted average of all possible values. For example, rolling a fair 6-sided die, the expected value is $\frac{1+2+3+4+5+6}{6}=3.5$, which is not a possible outcome of the die roll.

1. False.

• Corrected statement: In most applications, discrete random variables represent counted data, while continuous random variables represent measured data.

1. False.

• Explanation: The mean of a random variable describes the central tendency (average outcome), while the variance or standard deviation describes how outcomes vary.

1. Discrete random variable.

• Reasoning: Concert attendance is a countable number (you cannot have a fraction of a person in attendance).

1. Continuous random variable.

• Reasoning: The distance a baseball travels can take any value within a range (e.g., 325.7 feet, 410.2 feet) and is measured, not counted.

1. Discrete random variable.

• Reasoning: The number of cars is a countable value (you cannot have a fraction of a car in the parking lot).

1. Continuous random variable.

• Reasoning: Volume of blood is a measurable quantity that can take any value in a range (e.g., 5.2 mL, 7.8 mL).

1. Discrete random variable.

• Reasoning: The number of texts is a countable value (you cannot send a fraction of a text).