Random Reality Slides
Statistics Unit 5
Random Variables
Understanding Probability Distributions and Statistical Measurements
Learning Objectives
Distinguish
Differentiate between discrete and continuous random variables.
Graph
Construct and interpret discrete probability histograms.
Compute
Calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)).
Transform
Analyze the effects of linear functions on random variables.
Counting vs. Measuring
Discrete
The Result of Counting
- Number of defective light bulbs
- Students in a classroom
- Tuition cost (Dollars/Cents)
Continuous
The Result of Measurement
- Air pressure in a tire
- Daily rainfall in inches
- Time to register for classes
The Boredom Tolerance Test
Study of 20,000 subjects (Ages 25–35)
TABLE 5-2
| Score \(x\) | Prob \(P(x)\) |
|---|
| 0 | 0.07 |
| 1 | 0.13 |
| 2 | 0.18 |
| 3 | 0.30 |
| 4 | 0.22 |
| 5 | 0.08 |
| 6 | 0.02 |
| Total | 1.00 |
Histogram Rule
In a probability histogram, the Area of each bar equals the Probability.
Area = Height × 1
Practice Q
Probability of scoring 5 OR 6?
0.08 + 0.02 = 0.10
Expected Value \(\mu\)
The Formula
\[\mu = \sum x P(x)\]
"The balance point of the entire distribution."
Long-Term Average
The outcome you expect if the experiment is repeated thousands of times.
Center of Gravity
It is the theoretical 'average' value of the random variable \(x\).
Case Study: Infomercials
National Infomercial Marketing Association Study: Number of times buyers watched a product before purchasing.
| \(x\) (Viewings) | \(P(x)\) | \(x P(x)\) | Note |
|---|
| 1 | 0.27 | 0.27 | \(1 \times 0.27\) |
| 2 | 0.31 | 0.62 | \(2 \times 0.31\) |
| 3 | 0.18 | 0.54 | \(3 \times 0.18\) |
| 4 | 0.09 | 0.36 | \(4 \times 0.09\) |
| 5 | 0.15 | 0.75 | \(5 \times 0.15\) |
| Sum \(\sum x P(x) =\) | 2.54 | Expected Value | |
Meaning: On average, a customer watches the infomercial 2.54 times before buying.
Standard Deviation \(\sigma\)
Measuring the Spread and Uncertainty
Formula
\[\sigma = \sqrt{\sum (x - \mu)^2 P(x)}\]
Wait! Before finding \(\sigma\), you MUST find the mean \(\mu\) first.
Consistency
A Small \(\sigma\) means values are close to the mean. High predictability.
Risk
A Large \(\sigma\) means high variation. More risk and uncertainty.
Linear Transformations
Shifting and Scaling
Rule: \(L = a + bx\)
New Mean \(\mu_L = a + b \mu_x\)
New Spread \(\sigma_L = |b| \sigma_x\)
Adding (\(a\))
Adding a constant only shifts the center. It does not stretch or compress the data, so spread stays the same.
Multiplying (\(b\))
Multiplying scales everything. It changes both where the center is and how spread out the values are.
Carnival Challenge
Expected Earnings
COST TO PLAY: $2.00
Flip 3 fair coins. You win $1.00 for every Head. What are your long-run expected earnings?
| Heads \(x\) | Prob \(P(x)\) |
|---|
| 0 | 0.125 |
| 1 | 0.375 |
| 2 | 0.375 |
| 3 | 0.125 |
Tasks
Verdict:
If \(\mu < 2.00\), the house makes money on every player in the long run. If \(\mu > 2.00\), you've found a gold mine!
Big Picture
Identity
Know if your data is countable or measurable.
Location
The Mean (\(\mu\)) tells you where the center is.
Spread
Std Dev (\(\sigma\)) tells you how far it stretches.
Ready for Unit 5.2
Random Reality Worksheet
Random Reality
Unit 5: Probability Distributions
Name:
Date:
1
Discrete vs. Continuous
Identify each of the following random variables as Discrete (D) or Continuous (C).
___
Number of errors on a test
___
Amount of sugar in an apple (grams)
___
Temperature of a freezer (Celsius)
___
Number of text messages sent in a day
2
The Boredom Tolerance Test
A study of 20,000 subjects yielded the following probability distribution for test scores \(x\):
| Score \(x\) | \(P(x)\) |
|---|
| 0 | 0.07 |
| 1 | 0.13 |
| 2 | 0.18 |
| 3 | 0.30 |
| 4 | 0.22 |
| 5 | 0.08 |
| 6 | 0.02 |
A. Confirm this is a valid probability distribution. Explain why.
B. What is the probability that a subject scores at least a 4? Show your calculation.
3
Expected Value Challenge
Find the expected value \(\mu\) for the infomercial viewing study. Use the table below to organize your work.
| Viewings \(x\) | Probability \(P(x)\) | Calculation: \(x \times P(x)\) |
|---|
| 1 | 0.27 | |
| 2 | 0.31 | |
| 3 | 0.18 | |
| 4 | 0.09 | |
| 5 | 0.15 | |
| Sum \(\mu = \sum x P(x)\) | | |
Interpretation: On average, how many times does a customer view the ad before buying?
4
The Coin Flip Game
You pay $2.00 to flip 3 coins. You win $1.00 per Head. The probabilities for \(x\) Heads are: \(P(0)=0.125\), \(P(1)=0.375\), \(P(2)=0.375\), \(P(3)=0.125\).
Task: Show Your Work
Calculate the Expected Value (\(\mu\)) of your winnings below:
Random Reality Teacher Guide
Random Reality
Teacher Solution Key & Guide
Unit 5.1 - 5.2
Worksheet Answer Key
1. Discrete vs. Continuous
- D Number of errors on a test
- C Amount of sugar in an apple
- C Temperature of a freezer
- D Number of text messages sent
2. The Boredom Tolerance Test
A. Valid Distribution: Yes. (1) Each probability \(0 \le P(x) \le 1\) and (2) \(\sum P(x) = 1.00\).
B. Score at least 4: \(P(4) + P(5) + P(6) = 0.22 + 0.08 + 0.02 = \mathbf{0.32}\).
3. Infomercial Study Expected Value
| \(x\) | \(P(x)\) | \(x P(x)\) |
|---|
| 1 | 0.27 | 0.27 |
| 2 | 0.31 | 0.62 |
| 3 | 0.18 | 0.54 |
| 4 | 0.09 | 0.36 |
| 5 | 0.15 | 0.75 |
| Sum | 2.54 | |
Interpretation: A buyer watches the infomercial 2.54 times on average.
4. The Coin Flip Game
\(\mu = (0 \times 0.125) + (1 \times 0.375) + (2 \times 0.375) + (3 \times 0.125) = 0 + 0.375 + 0.75 + 0.375 = \mathbf{1.50}\).
Conclusion: You should not play. The expected winnings (\(\$1.50\)) are less than the cost to play (\(\$2.00\)). You lose \(\$0.50\) per game on average.
Instructional Guide
Common Misconceptions
- Students may confuse discrete vs. continuous. Remind them: Can you have 2.5 of them? If yes, it's continuous (money is a special case: discrete in cents, but treated as continuous in financial modeling).
- The Expected Value does NOT have to be a possible outcome (e.g., 2.54 viewings is impossible for one person, but it represents the long-term average).
- Shifting (\(a\)) vs. Scaling (\(b\)): Students often forget that adding doesn't change standard deviation. Use the "elevator" analogy: if everyone in an elevator gets 5 inches taller, the gap between the shortest and tallest person stays the same.
Teaching Strategy
The Balance Point: Use a physical ruler and coins. Place a coin at "0" and two coins at "3". Find where your finger must go to balance the ruler. This is \(\mu\).
