Topics 4.7–4.8: Random Variables & Probability Distributions

Key Vocabulary

Random Variable	A numerical outcome of random behavior (labeled with capital letters like X, Y)
Discrete Random Variable	Can only take a countable number of values (with space between values on a number line)
Continuous Random Variable	Can take an infinite number of values in an interval (no space between values)
Probability Distribution	A table, graph, or function showing all possible values and their probabilities
Expected Value (Mean)	The long-run average value of a random variable: μ_X = Σx_i·P(x_i)
Standard Deviation	Typical deviation from the mean: σ_X = √[Σ(x_i - μ_X)²·P(x_i)]

Video 1: Introduction to Random Variables (Topic 4.7)

[Video: ~8 min]

1. are numerical outcomes of .

2. Random variables are always labeled with a letter (like X, W, Y, or L).

3. A random variable can only take a number of values. There is between each value on the number line.

4. A random variable can take an number of values in an interval on the number line.

5. Classify each variable:

X = number of children in a household:

W = time in minutes to run a mile:

Y = number of dogs with location chip out of 10:

L = length of a person's index finger (cm):

Thermostat Example: The City of Austin recommends setting thermostats to 78 degrees. Let X = the number of degrees a thermostat is set below the recommended 78 degrees.

6. A displays the entire set of values of a random variable with their associated .

Properties of a Probability Distribution:

All probabilities are between and inclusive
The sum of all probabilities must equal

Thermostat Probability Distribution:

Temp	78	77	76	75	74	73	72
X	0	1	2	3	4	5	6
P(X)	0.03	0.03	0.05	0.18	0.31	0.28	0.12

7. Using the thermostat distribution, find P(X ≥ 4) = P(74 degrees or lower):
P(X ≥ 4) = + + =

Video 2: Describing Probability Distributions (Topic 4.7)

[Video: ~5 min]

Prairie Dog Example: Prairie dogs are a keystone species. They mate once a year, and litter size (X = number of pups) varies.

X (pups)	1	2	3	4	5	6	7
P(X)	0.15	0.38	0.27	0.11	0.05	0.03	0.01

8. To describe a probability distribution, we must talk about , , and .

9. Describe the prairie dog pup distribution:

Shape:

Center (median): approximately pups

Spread (range): pups

10. Should a zoologist be surprised if a prairie dog had 6 or 7 pups?
Calculate: P(X = 6 or X = 7) = (relatively unlikely)

11. Describe the thermostat distribution (from Video 1):

Shape:

Center: median temperature is about degrees (X = )

Spread (range): degrees

Topic 4.8: Mean and Standard Deviation of Random Variables

[Video: ~7 min]

Formulas for Parameters of Discrete Random Variables

Mean (Expected Value): μ_X = Σx_i · P(x_i)

Standard Deviation: σ_X = √[Σ(x_i - μ_X)² · P(x_i)]

Variance: σ_X² = Σ(x_i - μ_X)² · P(x_i)

12. A is a numerical value measuring a characteristic of a population or distribution. It is a single, value.

13. The mean formula takes into account the different of each X value by multiplying each value by its .

14. Calculate the mean for prairie dog pups:
μ_X = 1(0.15) + 2(0.38) + 3(0.27) + 4(0.11) + 5(0.05) + 6(0.03) + 7(0.01) = pups

15. Interpret the mean: In the , if many prairie dog litters are randomly selected, the number of pups per litter will be about 2.66 pups.

16. The standard deviation for prairie dog pups is calculated to be σ_X = pups.

17. Interpret the standard deviation: The number of prairie dog pups in randomly selected litters will generally from the mean of 2.66 pups by about pups.

Insurance Example: A $25,000 renter's insurance policy costs $150/year.

Theft/vandalism: $3,000 payout, probability 0.0097
Fire (total loss): $25,000 payout, probability 0.0003
No claim: 99% of the time

Outcome	No Claim	Theft	Fire
X (Profit)	$150	-$2,850	-$24,850
P(X)	0.99	0.0097	0.0003

18. Calculate expected profit:
μ_X = 150(0.99) + (-2850)(0.0097) + (-24850)(0.0003) = $

19. Interpret: The insurance company can expect to make, on average, about $ per renter's policy from a large number of randomly selected policies.

Reflection Questions

[AI-Graded]

R1. Explain the difference between a discrete and continuous random variable. Give one original example of each (not from the video) and explain why it fits that category.

R2. A game show offers a contestant three doors. Behind one door is $10,000, behind another is $1,000, and behind the third is $0. The contestant picks randomly. Calculate the expected winnings and explain what this value means. Would you recommend playing this game if there's a $3,000 entry fee? Why or why not?

Exit Ticket

A carnival game costs $2 to play. You roll a fair six-sided die. If you roll a 6, you win $10. If you roll a 1, you win $3. Otherwise, you win nothing.

Define the random variable X as the player's NET profit (winnings minus cost)
Create a probability distribution for X
Calculate the expected value μ_X
Interpret your answer and determine if this is a "fair" game

Topics 4.7–4.8: Random Variables & Probability Distributions

Key Vocabulary

Video 1: Introduction to Random Variables (Topic 4.7)

Video 2: Describing Probability Distributions (Topic 4.7)

Topic 4.8: Mean and Standard Deviation of Random Variables

Formulas for Parameters of Discrete Random Variables

Reflection Questions

Exit Ticket

Class Responses (Esc to close)