Chapter 2 study guide

Probability,
without the fog.

This chapter is about describing uncertainty with numbers. The ideas build in a straight line: define possible outcomes, give them probabilities, summarize what is typical, then measure how two uncertain things move together.

1Outcomes

→

2Probabilities

→

3Risk

→

4Relationships

The one-sentence version

A random variable turns an uncertain outcome into a number so we can calculate with it.

The starting point

Random variables

A random variable is a number whose value depends on chance. “Random” means we do not know the result yet—not that the result is chaotic.

Random

The value can change from one observation to the next.

Your next exam grade
Commute time tomorrow
Number of Wi-Fi failures in class
Whether it rains, coded 0 or 1

Not random

The value is already fixed by the situation.

The scheduled number of midterms in your course

Homework Q1 answer

Special case

Binary and Bernoulli variables

A binary variable has only two possible values, usually 0 and 1. A single 0-or-1 trial is called a Bernoulli random variable.

Next person is male
P(X = 0) = 0.45

Next person is female
P(X = 1) = 0.55

Key idea: binary variables are discrete, not continuous.

Organizing uncertainty

Probability distributions

A probability distribution lists every possible value and how likely it is. All probabilities must add to 1.

Worked example

Number of job applicants

Total = 1.00

Applicants, x	0	1	2	3	4
P(X = x)	0.45	0.30	0.15	0.08	0.02
F(x) = P(X ≤ x)	0.45	0.75	0.90	0.98	1.00

CDF, in plain English

The cumulative distribution function keeps a running total. At x = 2, it asks: “What is the chance of 0, 1, or 2 applicants?”

F(2) = 0.45 + 0.30 + 0.15 = 0.90

At least 20.250.15 + 0.08 + 0.02

At most 30.98Read F(3)

3 or 40.100.08 + 0.02

Continuous variables

Probability is area

For a continuous variable, the height of the curve is a density. Probability is the area under the curve over the range you care about.

0.20< 15 min

0.5815–20 min

0.22> 20 min

P(T < 20) = 0.20 + 0.58 = 0.78P(T < 15 or T > 20) = 0.42

Summarizing a distribution

Center, spread, and shape

Four words do most of the work: expected value, variance, standard deviation, and skewness.

E(X)

Expected value

The probability-weighted average—the long-run “center.”

E(X) = Σ x · P(X=x)

σ²

Variance

Average squared distance from the mean. It measures spread.

Var(X) = Σ p(x − μ)²

Standard deviation

The square root of variance. It is easier to interpret because it uses the original units.

SD(X) = √Var(X)

↝

Skewness

Measures asymmetry: which side has the longer tail.

Kurtosis describes tail thickness—not symmetry.

Worked example

Comparing total risk: Stock A vs. B

Higher SD = more risk

Economy	Probability	Stock A	Stock B
Boom	0.50	0.25	0.10
Normal	0.30	0.20	0.30
Recession	0.20	0.06	0.50

Stock A

Expected return 0.197

Variance 0.005161

Standard deviation 0.0718

Stock B

Expected return 0.24

Variance 0.0244

Standard deviation 0.1562

More total risk

Why? B’s returns are spread farther from its average. Its standard deviation is more than twice A’s.

Moving together

Covariance

Covariance tells you whether two variables tend to move in the same direction or opposite directions.

Positive↗ ↗

Usually move together

Zero↗ ·

No linear co-movement

Negative↗ ↘

Usually move opposite ways

Do not judge strength from covariance’s size.

Covariance changes when units change. Its sign is useful; its raw magnitude is not easy to compare.

TrueIndependent variables have covariance zero.
TrueCovariance carries the product of X’s and Y’s units.
FalseCovariance is unit-free.

A standardized relationship

Correlation

Correlation rescales covariance to a clean range from −1 to +1. Because it has no units, both sign and size are meaningful.

-10+1

Strong opposite movementNo linear relationshipStrong same-direction movement

ρ_XY =

Cov(X,Y)

SD(X) × SD(Y)

Full worked example

Assets X and Y

Strong negative link

State	p	Return X	Return Y
Boom	0.10	0.30	0.04
Normal	0.40	0.35	0.10
Recession	0.50	0.05	0.60

Find the means
E(X) = 0.195
E(Y) = 0.344

Find the variances
Var(X) = 0.021225
Var(Y) = 0.065824

Multiply paired deviations
Σp(x−μ_x)(y−μ_y)

Read the result
Cov(X,Y) = −0.03688
ρ = −0.99

Interpretation: when X is above its average, Y is usually below its average, and vice versa. The relationship is very strong.

The bell curve

The normal distribution

A normal distribution is described by its mean μ (the center) and variance σ² (the spread squared). Standardizing turns any normal value into a z-score.

z =

y − μ

Reading N(μ, σ²)

In this course, the second number is the variance. For Y ~ N(6, 9), the mean is 6 and the standard deviation is √9 = 3.

The Φ function

Φ(z) means “area to the left”

Φ(−0.80)0.2119

Φ(1.86)0.9686

P(Z > 2.11)0.01741 − 0.9826

P(−1.96 ≤ Z ≤ 1.96)0.95000.9750 − 0.0250

Three normal-probability examples

Y ~ N(6, 9)

P(Y ≤ 4)

z = (4 − 6) / 3 = −0.6670.2525

Y ~ N(−4, 9)

P(Y > −3)

z = (−3 + 4) / 3 = 0.3330.3694

Y ~ N(−4, 4)

P(−5 ≤ Y ≤ −1)

z: −0.5 to 1.50.6247

Before the exam

One-page cheat sheet

Use this as your final pass.

Probability language

At most x: P(X ≤ x)
At least x: P(X ≥ x)
Continuous: a to b: F(b) − F(a)
Outside a to b: 1 − P(a ≤ X ≤ b)

Center & risk

Mean: Σpx
Variance: Σp(x − μ)²
Standard deviation: √variance
More SD: More total risk

Relationships

Covariance sign: Direction
Correlation sign: Direction
|Correlation|: Strength
Independent: Covariance = 0

Normal curve

N(μ, σ²): Mean, variance
z-score: (y − μ) / σ
Φ(z): Area to the left
Right tail: 1 − Φ(z)

Remember

Draw the probability you want before calculating.

Most mistakes come from choosing the wrong side of the curve—not from the arithmetic.

Probability,without the fog.