Module 4 Quick Reference Sheet

Discrete Probability Distributions • Print this for easy reference!

Random Variable: Numerical outcome of a random experiment

Discrete: Countable values (from counting)

Continuous: Any value in an interval (from measuring)

Notation: X = random variable, x = specific value

Two conditions must be met:

1. 0 ≤ P(X = x) ≤ 1 for all x

2. Σ P(X = x) = 1

E(X) = μ = Σ [x · P(X = x)]

Interpretation: Long-run average value

Var(X) = σ² = Σ [(x - μ)² · P(X = x)]

σ = √Var(X)

Interpretation: Measures spread/variability

Four Conditions (FTIC):

Probability Formula:

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

where C(n,k) = n! / (k!(n-k)!)

Mean:

μ = np

Standard Deviation:

σ = √(np(1-p))

Use when:

np ≥ 10 AND n(1-p) ≥ 10

Then: X ~ N(μ = np, σ = √(np(1-p)))

Scenario	n	p	Example Question
Coin flips	# of flips	0.5 (fair coin)	"10 flips, find P(exactly 6 heads)"
Multiple choice guessing	# of questions	1/(# choices)	"20 questions, 4 choices each, find P(≥12 correct)"
Quality control	Sample size	Defect rate	"100 items, 3% defect rate, find P(≤2 defective)"
Medical treatment	# of patients	Success rate	"50 patients, 80% success rate, find μ and σ"

Note: For P(X ≥ k), use: 1 - P(X ≤ k-1)

Example 1: Expected Value

X: 1, 2, 3 with P(X=x): 0.2, 0.5, 0.3

E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1

Example 2: Binomial

Flip coin 5 times, find P(X=3 heads)

P(X=3) = C(5,3) × (0.5)³ × (0.5)² = 10 × 0.03125 = 0.3125