Probability Distributions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Random Variables
Definition:
A random variable is a variable whose value is determined by the outcome of a random event
• Usually denoted by capital letters: X, Y, Z
• Specific values denoted by lowercase: x, y, z
• Associates numerical values with outcomes of random experiments
2. Discrete vs Continuous Random Variables
Discrete Random Variables:
Can take on only countable values (specific, separate values)
Examples:
• Number of heads when flipping 3 coins (0, 1, 2, 3)
• Number of students in a class (20, 21, 22, ...)
• Outcome of rolling a die (1, 2, 3, 4, 5, 6)
• Key: You can COUNT them
Continuous Random Variables:
Can take on any value within a range (infinite possible values)
Examples:
• Height of students (165.3 cm, 170.8 cm, ...)
• Time to complete a task (12.5 min, 15.7 min, ...)
• Temperature (23.4°C, 25.6°C, ...)
• Key: You MEASURE them
Comparison:
| Aspect | Discrete | Continuous |
|---|---|---|
| Values | Countable, specific | Uncountable, any in range |
| Method | Count | Measure |
| Distribution | Probability Mass Function (PMF) | Probability Density Function (PDF) |
| Graph | Bar graph/Histogram | Smooth curve |
3. Discrete Probability Distribution
Definition:
A discrete probability distribution lists all possible values of a discrete random variable along with their probabilities
Requirements:
1. Each probability must be between 0 and 1: \( 0 \leq P(x) \leq 1 \)
2. Sum of all probabilities equals 1: \( \sum P(x) = 1 \)
Example Distribution:
Number of Heads when flipping 2 coins:
| X (# of heads) | P(X) |
|---|---|
| 0 | 0.25 |
| 1 | 0.50 |
| 2 | 0.25 |
Check: \( 0.25 + 0.50 + 0.25 = 1.00 \) ✓
4. Expected Value (Mean)
Definition:
The expected value (or mean) is the average value you would expect if you repeated an experiment many times
\[ E(X) = \mu = \sum [x \cdot P(x)] \]
Multiply each value by its probability, then sum all products
Example:
Find expected value for coin flip example above:
\( E(X) = 0(0.25) + 1(0.50) + 2(0.25) \)
\( E(X) = 0 + 0.50 + 0.50 \)
\( E(X) = 1 \) head (average)
5. Variance
Definition:
Variance measures how spread out the values are from the expected value
\[ \text{Var}(X) = \sigma^2 = \sum [(x - \mu)^2 \cdot P(x)] \]
\[ \text{Or: } \sigma^2 = E(X^2) - [E(X)]^2 \]
where \( E(X^2) = \sum [x^2 \cdot P(x)] \)
Example:
Find variance for coin flip (E(X) = 1):
Method 1: \( \text{Var}(X) = (0-1)^2(0.25) + (1-1)^2(0.50) + (2-1)^2(0.25) \)
\( = 1(0.25) + 0(0.50) + 1(0.25) = 0.50 \)
Or
Method 2: \( E(X^2) = 0^2(0.25) + 1^2(0.50) + 2^2(0.25) = 1.50 \)
\( \text{Var}(X) = 1.50 - (1)^2 = 0.50 \)
6. Standard Deviation
Definition:
Standard deviation is the square root of variance (same units as the data)
\[ \sigma = \sqrt{\text{Var}(X)} = \sqrt{\sigma^2} \]
• Measures typical distance from the mean
• Smaller \( \sigma \) = data more clustered around mean
• Larger \( \sigma \) = data more spread out
Example:
Find standard deviation for coin flip (Var(X) = 0.50):
\( \sigma = \sqrt{0.50} \approx 0.707 \)
The typical deviation from the mean is about 0.707 heads
7. Games of Chance
Expected Value in Games:
Expected value tells us the average gain or loss per game over many plays
• If \( E(X) > 0 \): You expect to win money (favorable game)
• If \( E(X) = 0 \): Fair game (break even)
• If \( E(X) < 0 \): You expect to lose money (unfavorable)
Example:
A game costs $5 to play. You roll a die. If you roll 6, you win $20. Otherwise, you win nothing.
Net Winnings Distribution:
| Outcome | Net Win (X) | P(X) |
|---|---|---|
| Roll 6 | $15 | 1/6 |
| Roll 1-5 | -$5 | 5/6 |
\( E(X) = 15(\frac{1}{6}) + (-5)(\frac{5}{6}) \)
\( E(X) = 2.50 - 4.17 = -1.67 \)
Expected loss: $1.67 per game (unfavorable game)
8. Choose the Better Bet
Strategy:
Compare expected values of different games. Choose the one with the higher expected value
Decision Rules:
• Higher expected value = better bet
• Positive expected value = you should play
• Both negative = choose lesser loss (closer to 0)
Example:
Game A: E(X) = -$1.67
Game B: E(X) = -$0.50
Choose Game B (smaller expected loss)
9. Quick Reference Summary
Key Formulas:
Expected Value: \( E(X) = \sum [x \cdot P(x)] \)
Variance: \( \text{Var}(X) = \sum [(x-\mu)^2 \cdot P(x)] \) or \( E(X^2) - [E(X)]^2 \)
Standard Deviation: \( \sigma = \sqrt{\text{Var}(X)} \)
Requirements for Probability Distribution:
• \( 0 \leq P(x) \leq 1 \) for all x
• \( \sum P(x) = 1 \)
📚 Study Tips
✓ Discrete: count it (whole numbers); Continuous: measure it (any value)
✓ Expected value is long-run average, not what happens in one trial
✓ Variance measures spread; standard deviation is in same units as data
✓ For games: positive E(X) means favorable, negative means unfavorable
✓ Always verify sum of probabilities equals 1