Discrete and Continuous Random Variables

RANDOM
VARIABLES

1. Illustrates a random variable (discrete and continuous).

2. Distinguishes between a discrete and a continuous random variable.

3. Construct a probability distribution.

OBJECTIVES

ACTIVITY 1:
WHAT I KNOW?

Multiple Choice

What is the probability of drawing a king from a standard deck of 52 cards?

• 1/52

• 1/13

• 4/52

• 1/4

Multiple Choice

A coin is flipped once. What is the sample space?

• {Heads, Tails}

• {H}

• {T}

• {Heads, Tails, Edge}

Multiple Choice

Which of the following is an example of discrete data?

• The height of students in a class

• The number of students in a classroom

• The time it takes to finish a race

• The weight of a package

Multiple Choice

What type of data is represented by the amount of rainfall in a week?

• Qualitative

• Quantitative Continuous

• Quantitative Discrete

• Categorical

Multiple Choice

If the relative frequency of an event is 0.4, what is the probability of that event occurring?

• 40%

• 4%

• 0.004

• 0.04

Multiple Choice

Which of the following is a valid way to organize data?

• Using a pie chart for numerical data

• Creating a frequency table

• Plotting random values without a scale

• Listing only the highest value

Multiple Choice

A set of data contains the following values: 2,3,3,4,4,4,5,2, 3, 3, 4, 4, 4, 5,2,3,3,4,4,4,5. Which number is the mode?

• 2

• 3

• 4

• 5

Multiple Choice

The mean of the numbers 10,20,30,40,50,10, 20, 30, 40, 50,10,20,30,40,50 is:

• 25

• 30

• 35

• 40

Multiple Choice

Find the median of the data set: 8, 12, 15, 22, 7, 9, 10, 8, 12, 15, 22, 7, 9, 10, 8, 12, 15, 22, 7, 9, 10

• 9

• 10

• 12

• 15

Multiple Choice

If the data set is: 5,5,6,8,9,10,10,5, 5, 6, 8, 9, 10, 10,5,5,6,8,9,10,10, which of the following is correct?

• Mean = 8, Median = 8, Mode = 5 and 10

• Mean = 7, Median = 8, Mode = 10

• Mean = 7, Median = 7.5, Mode = 5 and 10

• Mean = 7.5, Median = 8, Mode = 10

Now that we've reviewed some foundational concepts such as measures of central tendency and probability, let's reflect on how these ideas can help us understand randomness in the world around us. Have you ever wondered why certain events happen with regularity while others seem completely unpredictable? Today, we will dive into a topic that explains how we can mathematically analyze and understand these uncertainties—Random Variables.

Have you ever rolled a dice or flipped a coin? How do you know the chances of getting a certain number or side? What if you had to predict the result of many rolls or flips? How can we figure out the most likely outcomes?

Coin Flip Challenge

Open Ended

Flip a coin 10 times and record the result of each flip in the table, writing "H" for heads and "T" for tails. After each flip, immediately note the outcome. Use this data to analyze the randomness of the results. Record the data on your notebook and input here your answer with the format "Flip number-Result". ( Example: 1-H, 2-T, 3-T, and so on)

Open Ended

How many heads (H) and tails (T) did you get? Are they approximately equal?

Open Ended

Do the results seem random, or do they look like they might follow a pattern?

Just like how we can’t predict the exact number of heads or tails, random variables help us understand and predict the likelihood of various outcomes in situations where uncertainty is involved. the results of the coin flips are examples of random variables, and they can be analyzed using probability to help predict what might happen in future flips or similar scenarios.

Variable

- a characteristic or attribute of a sample or population that changes or varies for different individuals or things.

Qualitative Variable

generates categorical data.

Quantitative Variable

generates numerical data.

Multiple Choice

The height of students in a classroom.

• Quantitative Variable

• Qualitative Variable

Multiple Choice

The color of cars in a parking lot.

• Quantitative Variable

• Qualitative Variable

Multiple Choice

The types of music genres people listen to.

• Quantitative Variable

• Qualitative Variable

Multiple Choice

The test scores of students in a math exam.

• Quantitative Variable

• Qualitative Variable

Discrete Random Variable

is a random variable whose set of all possible values are countable or infinitely countable. It can be represented as separate points on a number line.

The following are examples of discrete random variables:

·         the number of correct answers in a 5-item true or false quiz

·         the number of siblings of your classmates

·         the number of people in each country

Continuous Random Variable

is a random variable whose set of all possible values are not countable or infinite. It can be represented as an interval.

The following are examples of continuous random variables:

·         the height of each student in a class

·         the weight of each plane baggage

·         the waiting time before a person gets a taxi in a taxi stand

Multiple Choice

The number of apples picked from an orchard.

• Discrete random variable

• Continuous random variable

Multiple Choice

The volume of milk in a container.

• Discrete random variable

• Continuous random variable

Multiple Choice

The number of text messages sent in a day.

• Discrete random variable

• Continuous random variable

Multiple Choice

The time it takes for a runner to complete a marathon.

• Discrete random variable

• Continuous random variable

Open Ended

Provide one example each of a discrete random variable and a continuous random variable. Then, explain why each example is classified as discrete or continuous.

Now that you’ve explored both discrete and continuous random variables, you’re ready to analyze data in many different situations. Keep practicing, and the next steps in statistics will become even more exciting!

Probability Distribution for a Discrete Random Variable

• Probability is the chance of an event occurring. In this chapter we will apply our knowledge on probability experiment in constructing a probability distribution for a discrete random variable.

• Discrete probability distributions can be presented by using a graph, table or notation formula. Note that a discrete probability distribution,  must satisfy the following two requirements:

Example 1:

• Consider a random experiment of tossing a fair coin three times. In this scenario, the domain can be defined as the set of all possible outcomes of the experiment and the range of the random variable as the total number of tails that comes out after tossing a coin three times.

  Let X be the number of heads in the tossing of fair coin three times (the random variable). 

  The set of possible outcomes (domain) of the experiment is as follows:

  {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Example 2:

If a single die is rolled, construct a probability distribution for the values of the variable and corresponding probability.

Example 2:

• Solution:         

              The sample points of the sample space consist of the values (outcomes) of the variable (x) 1,2,3,4,5, and 6. The probability of these outcomes is ⅙. Hence the probability distribution is presented, as follows:

Fill in the Blank

On a certain 6-day period, an office canteen kept records of the number of customers it served each day from Monday to Saturday. The number of customers per day was represented by X, as follows. Construct a probability distribution for the number of customers of the office canteen for the 6-day period.

MONDAY: Pr(X)= 0.15
TUESDAY: Pr(X)= 0.21

WEDENSDAY: Pr(X)= ___

.

Fill in the Blank

On a certain 6-day period, an office canteen kept records of the number of customers it served each day from Monday to Saturday. The number of customers per day was represented by X, as follows. Construct a probability distribution for the number of customers of the office canteen for the 6-day period.

MONDAY: Pr(X)= 0.15
TUESDAY: Pr(X)= 0.21

WEDENSDAY: Pr(X)= 0.16
THURSDAY: Pr(X)=___

.

Fill in the Blank

On a certain 6-day period, an office canteen kept records of the number of customers it served each day from Monday to Saturday. The number of customers per day was represented by X, as follows. Construct a probability distribution for the number of customers of the office canteen for the 6-day period.

MONDAY: Pr(X)= 0.15
TUESDAY: Pr(X)= 0.21

WEDENSDAY: Pr(X)= 0.16
THURSDAY: Pr(X)=0.18

FRIDAY: Pr(X)=___

.

Fill in the Blank

On a certain 6-day period, an office canteen kept records of the number of customers it served each day from Monday to Saturday. The number of customers per day was represented by X, as follows. Construct a probability distribution for the number of customers of the office canteen for the 6-day period.

MONDAY: Pr(X)= 0.15
TUESDAY: Pr(X)= 0.21

WEDENSDAY: Pr(X)= 0.16
THURSDAY: Pr(X)=0.18

FRIDAY: Pr(X)=0.22

SATURDAY: Pr(X)=___

.

Fill in the Blank

On a certain 6-day period, an office canteen kept records of the number of customers it served each day from Monday to Saturday. The number of customers per day was represented by X, as follows. Construct a probability distribution for the number of customers of the office canteen for the 6-day period.

MONDAY: Pr(X)= 0.15
TUESDAY: Pr(X)= 0.21

WEDENSDAY: Pr(X)= 0.16
THURSDAY: Pr(X)=0.18

FRIDAY: Pr(X)=0.22

SATURDAY: Pr(X)=0.08

TOTAL Pr(X)=___

.

Great job! You've completed your first lesson on Random Variables and Probability Distribution for Discrete Random Variables. This foundation will help you tackle more advanced topics in statistics. Keep practicing and applying what you've learned to strengthen your understanding of probability!