Random Variables Presentation Slides
Statistics Unit 5
Random Variables
Understanding Probability Distributions and Statistical Measurements
Probability Theory
Section 5.1 - 5.2
Learning Objectives
Distinguish
Differentiate between discrete and continuous random variables in real-world scenarios.
Graph
Construct and interpret discrete probability histograms.
Compute
Calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)) of a probability distribution.
Transform
Analyze the effects of linear functions on random variables.
What is a Random Variable?
Definition
A quantitative variable \(x\) that is measured or observed in an experiment.
It is called "random" because the value \(x\) takes on is a chance or random outcome.
Example: The number of eggs in a nest, or the weight of a polar bear cub.
x
Theoretical Outcome
Discrete Random Variables
The Result of Counting
Definition
When the observations of a quantitative random variable can take on only a finite number of values or a countable number of values.
Key Concept: "Countable"
Think: 1, 2, 3... Even if it goes to infinity (like drills until oil is found), you can still count the steps.
Number of defective light bulbs
Students in a classroom
Dollars/Cents (Tuition cost)
Continuous Random Variables
The Result of Measurement
Definition
When the observations can take on any of the countless number of values in a line interval.
Key Concept: "No Gaps"
Values like 20.126, 20.12678, etc. are possible. Temperature, height, and time are continuous.
Air pressure in a tire
Daily rainfall in inches
Time to register for classes
Quick Check
Identify each variable as Discrete or Continuous
1. Amount of gasoline needed to drive 200 miles.
Continuous
2. Number of bad checks drawn on a bank in one day.
Discrete
3. Height of students in your statistics class.
Continuous
4. Voters who voted in the last county election.
Discrete
Probability Distributions
An assignment of probabilities to the specific values or range of values of the random variable.
1
Individual Probability
A probability is assigned to each value of the random variable \(x\).
2
The Sum Rule
The sum of these probabilities must be exactly 1.
\[\sum P(x) = 1\]
Mutually Exclusive
In a discrete distribution, each outcome is separate. You cannot have two scores at the same time.
The Boredom Tolerance Test
Study of 20,000 subjects (Ages 25–35)
Table 5-2
| Score \(x\) | Probability \(P(x)\) |
|---|
| 0 | 0.07 |
| 1 | 0.13 |
| 2 | 0.18 |
| 3 | 0.30 |
| 4 | 0.22 |
| 5 | 0.08 |
| 6 | 0.02 |
| Total \(\sum P(x)\) | 1.00 |
Observation
The probability of a subject scoring a 3 is 0.30. This was calculated by taking 6,000 subjects divided by the total 20,000.
Mutually Exclusive Rule
What is the probability of scoring a 5 OR 6?
0.08 + 0.02 = 0.10
Visualizing Distributions
The Probability Histogram
.30.20.100
0
1
2
3
4
5
6
Width is Unity
Each bar is one unit wide.
Area = Probability
The height represents the probability. Because width = 1, the area equals the height.
Total Area = 1
The sum of the areas of all bars must equal 1.
The Mean \(\mu\)
Also Known As
Expected Value
Formula
\(\mu = \sum x P(x)\)
"Multiply each value by its probability and sum them up."
The mean acts as the balance point or the center of gravity of the distribution.
It represents the long-term average outcome if the experiment were repeated thousands of times.
Standard Deviation \(\sigma\)
Measuring the Spread and Risk
Variation Formula
\(\sigma = \sqrt{\sum (x - \mu)^2 P(x)}\)
"This measures how much the values of \(x\) tend to vary from the mean \(\mu\)."
Small \(\sigma\): High consistency, low risk.
Large \(\sigma\): High variation, high uncertainty.
Wait!
Before finding \(\sigma\), you MUST find \(\mu\) first.
Linear Transformations
Shifting and Scaling
What happens when we apply a rule like \(L = a + bx\) to our random variable?
New Mean \(\mu_L = a + b\mu_x\)
New Std Dev \(\sigma_L = |b|\sigma_x\)
The "Shift" (a)
Adding a constant only shifts the mean. It does not change the spread (\(\sigma\)).
The "Scale" (b)
Multiplying by a constant scales both the mean and the standard deviation.
Guided Practice
30:00 - 45:00
Suppose a discrete random variable \(x\) has the following distribution:
<table class="w-full text-3xl border border-slate-200 rounded-2xl overflow-hidden shadow-sm"><tbody><tr class="bg-slate-50 border-b border-slate-200"><td class="p-6 font-bold text-slate-400 border-r border-slate-200 text-center">\(x\)</td><td class="p-6 text-center font-bold">1</td><td class="p-6 text-center font-bold">2</td><td class="p-6 text-center font-bold">3</td></tr><tr><td class="p-6 font-bold text-slate-400 border-r border-slate-200 text-center">\(P(x)\)</td><td class="p-6 text-center">0.2</td><td class="p-6 text-center">0.5</td><td class="p-6 text-center">0.3</td></tr></tbody></table>
Work through these:
- Confirm \(\sum P(x) = 1\).
- Calculate \(\mu = \sum xP(x)\).
- Graph the probability histogram.
Calculation Step: \(\mu\)
1(0.2) = 0.2
2(0.5) = 1.0
3(0.3) = 0.9
Total (\(\mu\)) = 2.1
The expected outcome is slightly higher than 2. This makes sense as 2 is the most likely score (0.5).
Wrap-Up
Discrete vs Continuous
Can you count it? (Discrete). Must you measure it? (Continuous).
Graphs & Rules
Sum must be 1. Histograms use area to show probability.
\(\mu\) and \(\sigma\)
Mean is the weighted average. Std Dev is the spread.
Ready for the Problem Set?
Random Variables Practice Worksheet
Data Dynamics Worksheet
Topic: Discrete & Continuous Random Variables
Name:
Date:
Part 1: Classification
1. Identify whether the following random variables are discrete or continuous:
Height of a sunflower in a garden
Number of students enrolled in a statistics course
The exact volume of water in a 16-oz bottle
2. Classification: Which of the following are discrete and which are continuous?
a.
Speed of an airplane
b.
Age of a college professor chosen at random
c.
Number of books in the college bookstore
d.
Weight of a football player chosen at random
e.
Number of lightning strikes in Rocky Mountain National Park on a given day
Part 2: Probability Distributions
3. Consider each distribution. Determine if it is a valid probability distribution or not, and explain your answer.
(a)
<table class="w-full border-collapse border border-slate-300 text-center"><tbody><tr class="bg-slate-50"><td class="border border-slate-300 p-2 font-mono font-bold">\(x\)</td><td class="border border-slate-300 p-2">0</td><td class="border border-slate-300 p-2">1</td><td class="border border-slate-300 p-2">2</td></tr><tr><td class="border border-slate-300 p-2 font-mono font-bold">\(P(x)\)</td><td class="border border-slate-300 p-2">0.25</td><td class="border border-slate-300 p-2">0.60</td><td class="border border-slate-300 p-2">0.15</td></tr></tbody></table>
Explanation:
(b)
<table class="w-full border-collapse border border-slate-300 text-center"><tbody><tr class="bg-slate-50"><td class="border border-slate-300 p-2 font-mono font-bold">\(x\)</td><td class="border border-slate-300 p-2">0</td><td class="border border-slate-300 p-2">1</td><td class="border border-slate-300 p-2">2</td></tr><tr><td class="border border-slate-300 p-2 font-mono font-bold">\(P(x)\)</td><td class="border border-slate-300 p-2">0.25</td><td class="border border-slate-300 p-2">0.60</td><td class="border border-slate-300 p-2">0.20</td></tr></tbody></table>
Explanation:
Part 3: Marketing Case Study - Age Distribution
4. Marketing: Age. A supermarket "super shopper" is defined as a shopper for whom at least 70% of items purchased were on sale or purchased with a coupon. The following table shows the age distribution of these shoppers.
| Age Range (years) | 18–28 | 29–39 | 40–50 | 51–61 | 62 & Over |
|---|
Random Variables Answer Key
Answer Key
Data Dynamics: Random Variables Practice
Teacher Resource
STAT-5.1
Part 1: Classification
1. Additional Examples:
- Height of a sunflower: Continuous
- Number of students: Discrete
- Exact volume of water: Continuous
2. Textbook Classification:
a. Speed of airplane: Continuous
b. Age of professor: Continuous
c. Number of books: Discrete
d. Weight of player: Continuous
e. Lightning strikes: Discrete
Part 2: Probability Distributions
3(a) Valid?
YES
Explanation: \(\sum P(x) = 0.25 + 0.60 + 0.15 = 1.00\). All probabilities are between 0 and 1 inclusive.
3(b) Valid?
NO
Explanation: \(\sum P(x) = 0.25 + 0.60 + 0.20 = 1.05\). The total sum exceeds 1.00.
Part 3: Marketing - Age
4a. Validity Check
Yes. Sum of percentages: \(7\% + 44\% + 24\% + 14\% + 11\% = 100\%\), which converts to a total probability of 1.00.
4c. Expected Age (\(\mu\))
\(\mu = \sum xP(x)\)
\(= 23(0.07) + 34(0.44) + 45(0.24) + 56(0.14) + 67(0.11)\)
\(= 1.61 + 14.96 + 10.8 + 7.84 + 7.37\)
42.58 years
4d. Standard Deviation (\(\sigma\))
\(\sigma = \sqrt{\sum (x - \mu)^2 P(x)}\)
12.31 years
Part 4: Marketing - Income
5a. Validity Check
Yes. The probabilities sum to \(0.21 + 0.14 + 0.22 + 0.15 + 0.20 + 0.08 = 1.00\).
5c. Expected Income (\(\mu\))
\(10(0.21) + 20(0.14) + 30(0.22) + 40(0.15) + 50(0.20) + 60(0.08)\)
\(= 2.1 + 2.8 + 6.6 + 6.0 + 10.0 + 4.8\)
$32.3k
5d. Standard Deviation (\(\sigma\))
Calculation:
- (10-32.3)²(0.21) ≈ 104.43
- (20-32.3)²(0.14) ≈ 21.18
- (30-32.3)²(0.22) ≈ 1.16
- (40-32.3)²(0.15) ≈ 8.89
- (50-32.3)²(0.20) ≈ 62.66
- (60-32.3)²(0.08) ≈ 61.38
16.12 ($k)
