Learning Module 6 of Quantitative Methods 11–14% of the exam

Statistical Distributions

8 lessons. About 54 minutes in cram mode, 248 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary Practice this module

Lessons

1
Random variables and distribution functions
interpret 6 min cram, 26 min full
2
Unconditional expected value, variance and standard deviation
calculate 6 min cram, 26 min full
3
Discrete distributions: uniform, binomial and Poisson
calculate 8 min cram, 36 min full
4
Continuous distributions: uniform, normal and logistic
interpret 6 min cram, 30 min full
5
Why asset prices are log-normal
calculate 8 min cram, 32 min full
6
Probability trees, conditional expectations and conditional variance
calculate 8 min cram, 36 min full
7
Covariance and correlation from a joint probability table
calculate 6 min cram, 30 min full
8
Bayes' theorem: updating a probability with new information
calculate 6 min cram, 32 min full

What the exam expects you to do

Calculate, interpret, and evaluate unconditional expected values for mean, variance, and covariance.Random variables and distribution functions
Calculate, interpret, and evaluate the principal moments of key statistical distributions used in finance.Discrete distributions: uniform, binomial and Poisson
Calculate, interpret, and evaluate conditional expectations, variances, and covariances.Probability trees, conditional expectations and conditional variance
Formulate investment problems through Bayesian updating.Bayes' theorem: updating a probability with new information

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Random variables and distribution functions

Known

MemorizeWhich function, which variable

Function	Gives	Discrete	Continuous
Mass (PMF)	$P(X = x)$	yes	no (every point has 0)
Density (PDF)	density: area = probability	no	yes
Cumulative (CDF)	$P(X \le x)$	staircase	smooth curve

Memory hook Mass sits on points, density spreads over a range, cumulative keeps a running total.

Know it cold Continuous: $P(a \le X \le b) = F(b) - F(a)$ . Discrete with both ends included: $F(b) - F(a - 1)$ .

MemorizeLaw of large numbers against the central limit theorem

Know it cold More trials: the sample mean converges to the true mean (law of large numbers). The shape of the sample mean: approximately normal (central limit theorem).

Key terms

random variable Use it
stochastic variable Know the idea
discrete random variable Use it
continuous random variable Use it
probability mass function Memorize
probability density function Memorize
cumulative distribution function Memorize
law of large numbers Use it

Two kinds of random variable and their functions

Random variablea number for each outcome of an uncertain event

Discrete Memorize countable outcomes, such as a count of defaults
- Probability mass function (PMF) P(X = x) for each outcome
- Cumulative distribution function (CDF) a staircase that jumps at each outcome
Continuous Memorize any value in a range, such as a return
- Probability density function (PDF) probability is the area under it
- Cumulative distribution function (CDF) a smooth curve

Formulas

Cumulative distribution function Memorize

F(x) = P(X \le x)

Quick example

A cumulative table reads 0.09, 0.25 and 0.69 at x = 0, 1 and 2. P(X ≤ 1) = F(1) = 0.25.

Probability of a range, continuous variable Memorize

P(a \le X \le b) = \int_a^b f(x)\,dx = F(b) - F(a)

Quick example

F(x) = x² on the range 0 to 1. P(0.2 ≤ X ≤ 0.5) = 0.25 − 0.04 = 0.21.

On your calculator

0.5x²−0.2x²= 0.21

x² square

0.5ENTER×0.2ENTER×− 0.21

ENTER enter (on the HP, puts the number on the stack)

Probability of a range, discrete variable Use it

P(a \le X \le b) = \sum_{x=a}^{b} P(X = x)

Quick example

Mass of 0.12, 0.38 and 0.28 at x = 1, 2 and 3. P(1 ≤ X ≤ 3) = 0.78.

On your calculator

0.12+0.38+0.28= 0.78

0.12ENTER0.38+0.28+ 0.78

ENTER enter (on the HP, puts the number on the stack)

Probabilities add to 1 Memorize

\sum_x P(X = x) = 1, \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1

Quick example

Four outcomes have probabilities 0.22, 0.41, 0.19 and p. p = 1 − 0.82 = 0.18.

On your calculator

1−0.22−0.41−0.19= 0.18

1ENTER0.22−0.41−0.19− 0.18

ENTER enter (on the HP, puts the number on the stack)

Density as the slope of the CDF Know the idea

f(x) = \dfrac{dF(x)}{dx}

Quick example

F(x) = x² gives f(x) = 2x, so the density at x = 0.5 is 1.0.

Unconditional expected value, variance and standard deviation

Known

MemorizeForecast formulas

Measure	Formula	Weights
Expected value	Σ P(x) × x	probabilities
Variance	Σ P(x) × (x − E[X])²	probabilities
Variance (shortcut)	E[X²] − (E[X])²	probabilities
Std. deviation	√variance	same units as x
Distance in σ	(x − E[X]) ÷ σ	standard deviations

Memory hook Same shapes as the historical formulas of Learning Module 5, with the weight 1/n (or 1/(n − 1)) swapped for the probability P.

Know it cold Forecast with probabilities: weight by P, never divide by n − 1. History: sample formulas with n − 1.

Key terms

unconditional mean Memorize
unconditional variance Memorize
unconditional standard deviation Memorize
probability distribution Know the idea
scenario analysis Know the idea

From a table of scenarios to risk and reward

Probability distribution of outcomeseach outcome with its probability; probabilities add to 1

Unconditional mean (expected value) Memorize Σ P × x, the probability-weighted average
- Reads as the forecast, not a promised outcome
Unconditional variance Memorize E[X²] − (E[X])², units are squared
- Weights probabilities, so no n − 1
Unconditional standard deviation Memorize √variance, same units as the outcomes
- Reads as the typical distance from the forecast

Formulas

Expected value Memorize

E[X] = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i

Quick example

A return is 11% with probability 0.4 and 5% with probability 0.6. E = 0.4 × 11 + 0.6 × 5 = 7.4%.

On your calculator

0.4×11+(0.6×5)= 7.4

0.4ENTER11×0.6ENTER5×+ 7.4

ENTER enter (on the HP, puts the number on the stack)

Variance of a forecast Memorize

\mathrm{Var}(X) = \sigma_X^2 = \sum_{i=1}^{n} P(x_i)\,[x_i - E[X]]^2

Quick example

Same forecast, E[X] = 7.4. 0.4 × (11 − 7.4)² + 0.6 × (5 − 7.4)² = 5.184 + 3.456 = 8.64 (%²).

On your calculator

(11−7.4)x²×0.4+((5−7.4)x²×0.6)= 8.64

x² square

11ENTER7.4−ENTER×0.4×5ENTER7.4−ENTER×0.6×+ 8.64

ENTER enter (on the HP, puts the number on the stack)

Variance shortcut Memorize

\mathrm{Var}(X) = E\big[(X - E[X])^2\big] = E[X^2] - (E[X])^2

Quick example

Returns 11% (0.4) and 5% (0.6). E[X] = 7.4, E[X²] = 0.4 × 121 + 0.6 × 25 = 63.4. Var = 63.4 − 54.76 = 8.64 (%²).

On your calculator

0.4×11x²+(0.6×5x²)=−7.4x²= 8.64

x² square

11ENTER×0.4×5ENTER×0.6×+7.4ENTER×− 8.64

ENTER enter (on the HP, puts the number on the stack)

Standard deviation of a forecast Memorize

\sigma_X = \sqrt{\mathrm{Var}(X)}

Quick example

Variance 8.64 (%²). √8.64 = 2.94%.

On your calculator

8.64√x 2.94

√x square root

8.64g√x 2.94

g blue shift key · √x square root

Distance from the mean in standard deviations Use it

\dfrac{x - E[X]}{\sigma_X}

Quick example

E[X] = 7.4%, σ = 2.94%. An outcome of 2%: (2 − 7.4) ÷ 2.94 = −1.84.

On your calculator

2−7.4=÷2.94= -1.8367

2ENTER7.4−2.94÷ -1.8367

ENTER enter (on the HP, puts the number on the stack)

Discrete distributions: uniform, binomial and Poisson

Known

MemorizeDiscrete distributions: the moments

Distribution	Mean	Variance	Skewness
Discrete uniform, a to b	$(a+b) \div 2$	$[(b-a+1)^2 - 1] \div 12$	0
Binomial (n, p)	$np$	$np(1-p)$	$(1-2p) \div \sqrt{np(1-p)}$
Poisson (λ)	$\lambda$	$\lambda$	$1 \div \sqrt{\lambda}$

Memory hook Binomial variance is the mean times the chance of failure: np × (1 − p). Poisson is the lazy one: one number, λ, is both the mean and the variance.

Know it cold Default by t: $1 - e^{-\lambda t}$ , never $\lambda t$ . Expected time to default: $1/\lambda$ . The normal approximation needs np and n(1 − p) above 5.

Key terms

discrete uniform distribution Memorize
Bernoulli trial Use it
binomial distribution Memorize
binomial stock-price tree Use it
normal approximation to the binomial Use it
Poisson distribution Memorize
default intensity Use it
hazard rate Know the idea

How the discrete distributions relate

Discrete distributions in financecountable outcomes with a known mean and variance

Discrete uniform Memorize every whole-number outcome equally likely
Bernoulli trial Use it one trial: success with probability p, failure 1 − p
- Binomial Memorize successes in n independent trials; mean np
  - Poissonn large, p small, np = λ; events at a constant rate

Formulas

Discrete uniform mean and variance Memorize

\mu = \dfrac{a+b}{2}, \quad \sigma^2 = \dfrac{(b-a+1)^2 - 1}{12}

Quick example

Whole numbers 0 to 4, all equally likely: mean 2, variance (25 − 1) ÷ 12 = 2.

Binomial probability Use it

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

Quick example

n = 5, p = 0.3: P(X = 2) = 10 × 0.09 × 0.343 = 0.3087.

On your calculator

52NDnCr2=×(0.3y^x2)×(0.7y^x3)= 0.3087

2ND second function (the shifted label above a key) · y^x power

5gn!2gn!÷3gn!÷0.3ENTER2y^x×0.7ENTER3y^x× 0.3087

g blue shift key · ENTER enter (on the HP, puts the number on the stack) · y^x power

Binomial mean and variance Memorize

\mu = np, \quad \sigma^2 = np(1-p)

Quick example

n = 36, p = 0.25: mean 9, variance 36 × 0.25 × 0.75 = 6.75.

Binomial price node Use it

S = S_0\,u^{i}\,d^{j}

Quick example

S₀ = 50, u = 1.2, d = 0.9, two ups and one down: 50 × 1.44 × 0.9 = 64.80.

On your calculator

50×1.2x²×0.9= 64.8

x² square

50ENTER1.2ENTER××0.9× 64.8

ENTER enter (on the HP, puts the number on the stack)

Normal approximation to the binomial Know the idea

X \approx N\big(np,\; np(1-p)\big) \text{ if } np > 5 \text{ and } n(1-p) > 5

Quick example

n = 100, p = 0.1: np = 10 and n(1 − p) = 90 are both above 5, so X is about N(10, 9).

Poisson probability Use it

P(k) = \dfrac{(\lambda t)^k e^{-\lambda t}}{k!}

Quick example

λ = 2 a day, one day: P(1) = 2 × e^(−2) = 0.2707.

On your calculator

2×2+|-2NDe^x= 0.2707

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

2ENTER2CHSge^x× 0.2707

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Poisson mean and variance Memorize

\mu = \sigma^2 = \lambda \quad \text{(one unit of time)}

Quick example

5 defaults a year on average: variance 5, standard deviation 2.24.

Probability of default by time t Memorize

P(\text{default by } t) = 1 - e^{-\lambda t}, \quad E[t_d] = \dfrac{1}{\lambda}

Quick example

λ = 1.5% a year, 6 years: 1 − e^(−0.09) = 8.61%. Expected time to default 1 ÷ 0.015 = 66.7 years.

On your calculator

0.015×6=+|-2NDe^x+|-+1= 0.0861

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.015ENTER6×CHSge^xCHS1+ 0.0861

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Continuous distributions: uniform, normal and logistic

Known

MemorizeContinuous distributions: the moments

Distribution	Mean	Variance	Kurtosis (excess)
Continuous uniform [a, b]	$(a+b)/2$	$(b-a)^2/12$	1.8 (excess −1.2)
Normal N(μ, σ²)	μ	σ²	3 (0)
Logistic (μ, s)	μ	$s^2\pi^2/3$	4.2 (1.2)

Memory hook Uniform 1.8, normal 3, logistic 4.2: flat, normal, fat. The 12 in both uniform variances is the same 12.

Know it cold Normal: 68% within ±1σ, about 95% within ±2σ. Logistic variance = $s^2\pi^2/3$ , so it scales with $s^2$ .

Key terms

continuous uniform distribution Memorize
inverse transform sampling Use it
normal distribution Memorize
standard normal distribution Use it
logistic distribution Memorize
location parameter Know the idea
scale parameter Know the idea
four moments of a distribution Use it

Continuous distributions by tail weight

Continuous distributionssmooth curves; probability is area

Continuous uniform Memorize flat, no tails; kurtosis 1.8
Normal Memorize symmetric bell; kurtosis 3
Logistic Memorize symmetric bell, heavier tails; kurtosis 4.2
Log-normal Use it next lesson: skewed right, bounded at zero

Formulas

Continuous uniform mean and variance Memorize

\mu = \dfrac{a+b}{2}, \quad \sigma^2 = \dfrac{(b-a)^2}{12}

Quick example

Uniform on [0, 12]: mean 6, variance 144 ÷ 12 = 12.

Continuous uniform probability Use it

P(x_1 \le X \le x_2) = \dfrac{x_2 - x_1}{b - a}

Quick example

Uniform on [10, 50]: P(20 ≤ X ≤ 30) = 10 ÷ 40 = 0.25.

On your calculator

30−20=÷(50−10)= 0.25

30ENTER20−50ENTER10−÷ 0.25

ENTER enter (on the HP, puts the number on the stack)

Normal ranges Memorize

P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.68, \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.95

Quick example

N(5%, 10%²): about 95% of outcomes lie between 5 − 20 = −15% and 5 + 20 = 25%.

On your calculator

5−(2×10)= -15
5+(2×10)= 25

5ENTER2ENTER10×− -15
5ENTER2ENTER10×+ 25

ENTER enter (on the HP, puts the number on the stack)

Logistic mean and variance Memorize

E[X] = \mu, \quad \sigma^2 = \dfrac{s^2 \pi^2}{3}

Quick example

Scale s = 1.5: 2.25 × 9.8696 ÷ 3 = 7.40, so the standard deviation is 2.72.

On your calculator

1.5x²×9.8696÷3= 7.4022
√x 2.7207

x² square · √x square root

1.5ENTER×9.8696×3÷ 7.4022
g√x 2.7207

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Why asset prices are log-normal

Known

MemorizeNormal versus log-normal

	Normal	Log-normal
Range	−∞ to +∞	0 to +∞
Shape	symmetric	skewed right
Models	log returns (continuously compounded)	prices
Parameters	its own mean and variance	mean and variance of $\ln Y$

Memory hook Read the name as an instruction: log it and it turns normal. Prices have a floor at zero and no ceiling, so the tail runs right.

Know it cold If $\ln Y$ is normal, $Y$ is log-normal. Mean of $Y$ is above its median.

MemorizeProjecting a log-normal price

Item	Formula
Log price	$\ln S_T = \ln S_0 + (\mu - \sigma^2/2)T$
Price	$S_T = S_0 e^{(\mu - \sigma^2/2)T}$
σ of log return over T	$\sigma\sqrt{T}$
Mean of $e^X$	$e^{\mu_X + \sigma_X^2/2}$

Memory hook Volatility drags the typical price down: subtract half the variance from the drift.

Know it cold Drop σ²/2 or add it and you land on the two wrong options.

Key terms

normal distribution Use it
log-normal distribution Memorize
log return Memorize
drift adjustment Memorize
geometric Brownian motion Know the idea
Wiener process Know the idea

From returns to prices

Price after T periods

S_T = S_0\,e^{r}

Log return Memorize normal, can be negative without limit
- Sum of period returns adds across time
- Normal the modeling assumption
Price Memorize log-normal: bounded at 0, skewed right
- Mean above median the long right tail pulls the mean up

Formulas

Price from a continuously compounded return Memorize

S_T = S_0\,e^{r_{0,T}}, \qquad r_{0,T} = \ln\left(\frac{S_T}{S_0}\right)

Quick example

A 25.00 share has a continuously compounded return of 8% over the year. $S_T = 25 \times e^{0.08}$ = 27.08.

On your calculator

0.082NDe^x×25= 27.08

2ND second function (the shifted label above a key) · e^x e to the power x

0.08ge^x25× 27.08

g blue shift key · e^x e to the power x

Continuously compounded returns add Memorize

r_{0,T} = r_{0,1} + r_{1,2} + \cdots + r_{T-1,T}

Quick example

A share goes 40.00, 42.40, 41.00, 44.00. Daily returns 5.83% − 3.36% + 7.06% = 9.53%, which equals $\ln(44 \div 40)$ .

On your calculator

44÷40=LN 0.0953, the same total in one step

LN natural logarithm

44ENTER40÷gLN 0.0953

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Projected log price Memorize

\ln S_T = \ln S_0 + \left(\mu - \tfrac{\sigma^2}{2}\right)T, \quad S_T = S_0\,e^{(\mu - \sigma^2/2)T}

Quick example

S₀ = 25, μ = 6%, σ = 16%, T = 3: 25 × e^((0.06 − 0.0128) × 3) = 25 × e^0.1416 = 28.80.

On your calculator

0.16x²÷2=+|-+0.06=×3=2NDe^x×25= 28.80

x² square · +|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.06ENTER0.16ENTER×2÷−3×ge^x25× 28.80

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Standard deviation of the log return over T Use it

\sigma_T = \sigma\sqrt{T}

Quick example

σ = 22% a year, T = 3: 22 × √3 = 38.11%.

On your calculator

3√x×22= 38.1051

√x square root

3g√x22× 38.1051

g blue shift key · √x square root

Mean of a log-normal variable Use it

E[Y] = \mu_Y = e^{\mu_X + \sigma_X^{2}/2}

Quick example

Annual log return mean 6%, standard deviation 30%, price 100. Mean price $100 \times e^{0.105}$ = 111.07, above the median 106.18.

On your calculator

0.3x²÷2+0.06=2NDe^x×100= 111.07

x² square · 2ND second function (the shifted label above a key) · e^x e to the power x

0.3ENTER×2÷0.06+ge^x100× 111.07

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Variance of a log-normal variable Use it

\sigma_Y^2 = \left(e^{\sigma_X^2} - 1\right)e^{2\mu_X + \sigma_X^2}

Quick example

μ_X = 0.06, σ_X = 0.30: (e^0.09 − 1) × e^0.21 = 0.094174 × 1.233678 = 0.1162, standard deviation 0.3409.

On your calculator

0.3x²=STO1 0.09
2NDe^x−1=STO2 0.094174
2×0.06+RCL1=2NDe^x×RCL2= 0.116181

x² square · STO store · 2ND second function (the shifted label above a key) · e^x e to the power x · RCL recall a stored number

0.3ENTER×STO1 0.09
ge^x1−STO2 0.094174
0.06ENTER2×RCL1+ge^xRCL2× 0.116181

ENTER enter (on the HP, puts the number on the stack) · STO store · g blue shift key · e^x e to the power x · RCL recall a stored number

Probability trees, conditional expectations and conditional variance

Known

MemorizeTree rules

Item	Rule
Branches from one node	probabilities add to 1
End-branch (joint) probability	product along the path: P(S) × P(x \| S)
All end branches	add to 1
Conditional E[X \| S]	weight outcomes by P(x \| S) inside branch S
Unconditional E[X]	Σ E[X \| Sᵢ] × P(Sᵢ)

Memory hook Fold the tree from the right: each first-stage node becomes one number, then weight those numbers by the first-stage probabilities.

Know it cold Total probability rule needs scenarios that are mutually exclusive and exhaustive, not equally likely.

Key terms

probability tree Use it
node Know the idea
conditional probability Use it
conditional expected value Memorize
total probability rule for expected value Memorize
conditional variance Use it
conditional covariance Know the idea

Reading a two-stage probability tree

Probability treethe first uncertainty, then the outcomes within each branch; also called a decision tree

Node Know the idea a point where uncertainty resolves
- Branch probabilities from one node they add to 1
End-branch (joint) probability Use it multiply along the path
- All end branches add to 1
Conditional expected value Memorize the forecast within one first-stage branch
- Unconditional E[X] weight each by its branch probability and add

Formulas

Joint probability along a path Use it

P(A \text{ and } B) = P(B \mid A)\,P(A)

Quick example

A scenario has probability 0.65, and an outcome within it has conditional probability 0.75. The joint probability is 0.65 × 0.75 = 0.4875.

On your calculator

0.65×0.75= 0.4875

0.65ENTER0.75× 0.4875

ENTER enter (on the HP, puts the number on the stack)

Conditional probability Use it

P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}, \quad P(B) \neq 0

Quick example

Joint probability 0.4875, P(B) = 0.65. P(A | B) = 0.4875 ÷ 0.65 = 0.75.

On your calculator

0.4875÷0.65= 0.75

0.4875ENTER0.65÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Conditional expected value Memorize

E[X \mid S] = P(x_1 \mid S)\,x_1 + P(x_2 \mid S)\,x_2 + \cdots + P(x_n \mid S)\,x_n

Quick example

Given scenario S, an outcome of 15 has probability 0.9 and an outcome of 5 has 0.1. E[X | S] = 13.5 + 0.5 = 14.0.

On your calculator

0.9×15+(0.1×5)= 14

0.9ENTER15×0.1ENTER5×+ 14

ENTER enter (on the HP, puts the number on the stack)

Total probability rule for expected value Memorize

E[X] = E[X \mid S_1]\,P(S_1) + E[X \mid S_2]\,P(S_2) + \cdots + E[X \mid S_n]\,P(S_n)

Quick example

E[X | S₁] = 5.0 with P(S₁) = 0.65, and E[X | S₂] = −2.0 with P(S₂) = 0.35. E[X] = 3.25 − 0.70 = 2.55.

On your calculator

0.65×5+(0.35×2+|-)= 2.55

+|- change sign

0.65ENTER5×0.35ENTER2CHS×+ 2.55

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Conditional variance Use it

\mathrm{Var}(X \mid S) = E\big[(X - E[X \mid S])^2 \mid S\big] = E[X^2 \mid S] - \big(E[X \mid S]\big)^2

Quick example

Given S, outcomes 5 and −1 each have probability 0.5, so E[X | S] = 2. Variance = 0.5 × 9 + 0.5 × 9 = 9. Shortcut: E[X² | S] = 13, and 13 − 2² = 9.

On your calculator

(5−2)x²×0.5+((1+|-−2)x²×0.5)= 9

x² square · +|- change sign

5ENTER2−ENTER×0.5×1CHSENTER2−ENTER×0.5×+ 9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Conditional covariance Know the idea

\mathrm{Cov}(X, Y \mid S) = E\big[(X - E[X \mid S])(Y - E[Y \mid S]) \mid S\big]

Quick example

Given S, X is 9 and Y is 11 with probability 0.75; X is 5 and Y is 3 with probability 0.25. E[X | S] = 8, E[Y | S] = 9. Cov = 0.75 × (1)(2) + 0.25 × (−3)(−6) = 6.0.

On your calculator

0.75×1×2+(0.25×3+|-×6+|-)= 6

+|- change sign

0.75ENTER1×2×0.25ENTER3CHS×6CHS×+ 6

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance and correlation from a joint probability table

Known

MemorizeReading the signs of the products

R_A vs mean	R_B vs mean	Product	Pushes covariance
above	above	+	up
below	below	+	up
above	below	−	down
below	above	−	down

Memory hook Each scenario casts a vote: same side of the means is a plus, opposite sides a minus, and the probability is the size of the vote.

Know it cold $\mathrm{Cov} = E[R_AR_B] - E[R_A]E[R_B]$ , or $\sum P \times \text{dev}_A \times \text{dev}_B$ . Independent: $P(X,Y) = P(X)P(Y)$ and Cov = 0.

Key terms

unconditional covariance Memorize
joint probability Use it
joint distribution function Know the idea
marginal probability Use it
contingency table Use it
scenario analysis Know the idea
independent random variables Use it
uncorrelated returns Use it

From a scenario table to correlation

Joint table: probability, R_A and R_B in each stateprobabilities add to 1; no n − 1

1. Expected returns Memorize E[R_A] = Σ P × R_A and E[R_B] = Σ P × R_B
2. Unconditional covariance Memorize E[R_A R_B] − E[R_A] E[R_B]
- Deviation form Σ P × (R_A − E_A) × (R_B − E_B)
3. Standard deviations Use it √ of Σ P × (R − E)²
4. Correlation Memorize Cov ÷ (σ_A σ_B), between −1 and +1

Independent, uncorrelated and dependent

Two random returnscovariance only sees straight-line links

Independent Use it P(X, Y) = P(X) P(Y); knowing one tells you nothing about the other
- Implies covariance 0, correlation 0, E[XY] = E[X]E[Y]
Uncorrelated Use it covariance 0 but a nonlinear link may remain
Positive covariance Memorize returns tend to sit on the same side of their means
Negative covariance Memorize one tends to be above while the other is below

Formulas

Expected return from scenarios Memorize

E[R] = \sum_{i} P(s_i)\, R_i

Quick example

Probabilities 0.25, 0.50, 0.25 with returns 12%, 6% and −4%. 0.25 × 12 + 0.50 × 6 + 0.25 × (−4) = 3 + 3 − 1 = 5%.

On your calculator

0.25×12+(0.5×6)+(0.25×4+|-)= 5

+|- change sign

0.25ENTER12×0.5ENTER6×+0.25ENTER4CHS×+ 5

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance shortcut Memorize

\sigma_{AB} = \mathrm{Cov}(R_A, R_B) = E[R_A R_B] - E[R_A]\,E[R_B]

Quick example

Two states, probabilities 0.6 and 0.4: A returns 10% and −5% (mean 4%), B returns 6% and 2% (mean 4.4%). E[R_A R_B] = 0.6 × 60 + 0.4 × (−10) = 32. Cov = 32 − 4 × 4.4 = 14.4 (%²).

On your calculator

0.6×60+(0.4×10+|-)=−(4×4.4)= 14.4

+|- change sign

0.6ENTER60×0.4ENTER10CHS×+4ENTER4.4×− 14.4

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance from a joint probability table Use it

\mathrm{Cov}(R_A, R_B) = \sum_{i} P(s_i)\,\big[R_{A,i} - E[R_A]\big]\big[R_{B,i} - E[R_B]\big]

Quick example

Two states with probabilities 0.6 and 0.4. A returns 10% and −5% (mean 4%); B returns 6% and 2% (mean 4.4%). 0.6 × (6)(1.6) + 0.4 × (−9)(−2.4) = 5.76 + 8.64 = 14.4 (%²).

On your calculator

0.6×(10−4)×(6−4.4)+(0.4×(5+|-−4)×(2−4.4))= 14.4

+|- change sign

10ENTER4−6ENTER4.4−×0.6×5CHSENTER4−2ENTER4.4−×0.4×+ 14.4

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Correlation from scenario returns Memorize

\rho_{XY} = \dfrac{\sigma_{XY}}{\sigma_X\,\sigma_Y} = \dfrac{\mathrm{Cov}(X, Y)}{\sigma_X\,\sigma_Y}

Quick example

Covariance −12, standard deviations 6% and 4%. −12 ÷ (6 × 4) = −0.50.

On your calculator

12+|-÷(6×4)= -0.5

+|- change sign

12CHSENTER6ENTER4×÷ -0.5

CHS change sign · ENTER enter (on the HP, puts the number on the stack)

Joint distribution function Know the idea

F_{12}(x_1, x_2) = P(X_1 \le x_1,\; X_2 \le x_2)

Quick example

In the grid below, P(R_A ≤ 7% and R_B ≤ 5%) adds the cells (7, 5), (7, −2), (−4, 5) and (−4, −2): 0.05 + 0 + 0.20 + 0.25 = 0.50.

On your calculator

0.05+0.20+0.25= 0.5

0.05ENTER0.20+0.25+ 0.5

ENTER enter (on the HP, puts the number on the stack)

Independence Use it

P(X, Y) = P(X)\,P(Y)

Quick example

P(X = 1) = 0.35 and P(Y = 1) = 0.60, with X and Y independent. P(X = 1 and Y = 1) = 0.35 × 0.60 = 0.21.

On your calculator

0.35×0.6= 0.21

0.35ENTER0.6× 0.21

ENTER enter (on the HP, puts the number on the stack)

Expected product of uncorrelated returns Use it

E[XY] = E[X]\,E[Y] \quad \text{if } X, Y \text{ are uncorrelated}

Quick example

Revenue is price times quantity. If expected price is 20 and expected quantity is 300 and they are uncorrelated, expected revenue is 20 × 300 = 6,000.

On your calculator

20×300= 6000

20ENTER300× 6000

ENTER enter (on the HP, puts the number on the stack)

Bayes' theorem: updating a probability with new information

Known

MemorizeBayes' update

Piece	Symbol	Meaning
Prior	P(A)	before the news
Likelihood	P(B \| A)	news if the event is true
Joint	P(B \| A) × P(A)	news and event
Marginal probability	P(B)	sum of every joint for the news
Posterior	P(A \| B)	joint ÷ P(B)

Memory hook The posterior is your share of all the ways the news could have happened: the event branch over every branch with the news.

Know it cold Posterior = likelihood × prior ÷ P(B). Posteriors over all scenarios add to 1.

MemorizeDiffuse prior

Memory hook No reason to favor any scenario, so give each the same prior. The priors then cancel and the likelihoods alone set the posterior.

Know it cold Equal priors: posterior = likelihood ÷ sum of the likelihoods.

Key terms

prior probability Memorize
likelihood Memorize
posterior probability Memorize
Bayes' theorem Memorize
marginal probability Memorize
joint probability Use it
total probability rule Use it
diffuse prior Use it

The four pieces of a Bayes update

Updating a probabilitynews arrives, so revise the view

Prior P(A) Memorize your probability before the news
Likelihood P(B | A) Memorize how likely the news is if the event is true
Marginal probability P(B) Memorize total chance of the news: add every branch that produces it
- Event branch P(B | A) × P(A)
- No-event branch P(B | not A) × P(not A)
Posterior P(A | B) Memorize event branch ÷ P(B)

Formulas

Bayes' theorem Memorize

P(A \mid B) = \dfrac{P(B \mid A)}{P(B)}\,P(A)

Quick example

P(A) = 0.35, P(B | A) = 0.85, P(B) = 0.50. P(A | B) = 0.85 × 0.35 ÷ 0.50 = 0.595.

On your calculator

0.85×0.35=÷0.5= 0.595

0.85ENTER0.35×0.5÷ 0.595

ENTER enter (on the HP, puts the number on the stack)

Marginal probability of the information Memorize

P(B) = P(B \mid A)\,P(A) + P(B \mid A^c)\,P(A^c)

Quick example

P(A) = 0.55, P(B | A) = 0.75, P(B | not A) = 0.35. P(B) = 0.4125 + 0.1575 = 0.57.

On your calculator

0.55×0.75+(0.45×0.35)= 0.57

0.55ENTER0.75×0.45ENTER0.35×+ 0.57

ENTER enter (on the HP, puts the number on the stack)

Missing likelihood from the marginal probability Use it

P(B \mid A^c) = \dfrac{P(B) - P(B \mid A)\,P(A)}{P(A^c)}

Quick example

P(B) = 0.34, P(B | A) = 0.55, P(A) = 0.25. P(B | not A) = (0.34 − 0.1375) ÷ 0.75 = 0.27.

On your calculator

0.34−(0.55×0.25)=÷0.75= 0.27

0.34ENTER0.55ENTER0.25×−0.75÷ 0.27

ENTER enter (on the HP, puts the number on the stack)

Bayes' theorem with several scenarios Use it

P(A_i \mid B) = \dfrac{P(B \mid A_i)\,P(A_i)}{\sum_{j} P(B \mid A_j)\,P(A_j)}

Quick example

Priors 0.55, 0.35, 0.10 and likelihoods 0.25, 0.45, 0.75. P(B) = 0.1375 + 0.1575 + 0.075 = 0.37. Posterior of the first: 0.1375 ÷ 0.37 = 0.372.

On your calculator

0.55×0.25=STO1
+(0.35×0.45)+(0.1×0.75)= 0.37
1/x×RCL1= 0.3716

STO store · 1/x reciprocal · RCL recall a stored number

0.55ENTER0.25×STO1
0.35ENTER0.45×+0.1ENTER0.75×+ 0.37
1/xRCL1× 0.3716

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number