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Exam cheatsheet

Skim it 10 minutes before the exam.

0 · Imports

import numpy as np
from scipy import stats
import math

1 · 8 functions

What	Python	MATLAB
Mean	np.mean(x)	mean(x)
Sample sd	np.std(x, ddof=1)	std(x)
Sample var	np.var(x, ddof=1)	var(x)
Normal CDF	stats.norm.cdf(x, mu, sig)	normcdf(x,mu,sig)
Normal quantile	stats.norm.ppf(p)	norminv(p)
t quantile	stats.t.ppf(p, df)	tinv(p,df)
Binomial CDF	stats.binom.cdf(k, n, p)	binocdf(k,n,p)
Poisson CDF	stats.poisson.cdf(k, lam)	poisscdf(k,lam)

Note ddof=1 — forget it and the whole question cascades. MATLAB defaults correct.

2 · 5 distributions

Dist	E	Var	Notes
Bin(n,p)	np	np(1−p)	P(X≥k)=1−cdf(k−1)
Poi(λ)	λ	λ	λ (equal) · E[Y²]=λ+λ²
U(a,b)	(a+b)/2	(b−a)²/12	quantile = a+p(b−a)
Exp(mean=m)	m	m²	F(t)=1−e^(−t/m)
N(μ,σ)	μ	σ²	norm.cdf / ppf
pdf f(x)=c/x²	integrate		integrate → F(t)=1−c/t

3 · 6 key formulas

Square of expectation

$E[X^{2}]=\operatorname{Var}(X)+\big(E[X]\big)^{2}$

for topics 8/11/17/20 (b)

CLT · sum

$S\sim N(n\mu,\ n\sigma^{2}),\quad \sigma_S=\sqrt{n}\,\sigma$

topics 5, 9

CLT · mean

$\bar X\sim N\Big(\mu,\ \frac{\sigma^{2}}{n}\Big),\quad \sigma_{\bar X}=\frac{\sigma}{\sqrt{n}}$

topics 10, 12, 14

t confidence interval

$\bar x\pm t_{\,n-1,\;\frac{1+\gamma}{2}}\cdot\frac{s}{\sqrt{n}}$

topics 4, 6, 7, 15

Proportion margin of error

$\text{ME}=z^{*}\sqrt{\frac{\hat p(1-\hat p)}{n}}$

topic 1 · normal ⟺ n·p̂(1−p̂) > 5

Estimators

$E\Big[\sum a_iX_i\Big]=\sum a_i\mu,\quad \operatorname{Var}=\sum a_i^{2}\sigma^{2}$

unbiased ⟺ Σaᵢ=1 · consistent ⟺ Var→0

4 · Solve-back templates

# critical value (sum, right tail α)
L = n*mu + stats.norm.ppf(1-alpha)*np.sqrt(n)*sigma
# critical value (mean, left tail α)
c = mu + stats.norm.ppf(alpha)*sigma/np.sqrt(n)      # z is negative
# sample size (one-sided)
z = stats.norm.ppf(alpha)
n = math.ceil( (z*sigma/(c-mu))**2 )                 # <- ceil, round up
# sample size (two-sided, total defect rate alpha)
z = stats.norm.ppf(1-alpha/2)                        # <- alpha/2 !
n = math.ceil( (z*sigma/half_width)**2 )

5 · Pre-submit checklist (each has cost marks)

Did you write ddof=1 ?
CI uses t.ppf not norm.ppf ? df = n−1 ? quantile (1+γ)/2 ?
"at least k" = 1 − cdf(k−1) ? (not 1 − cdf(k))
E[X²] = Var + mean² , not (E[X])² ?
CLT: sum uses √n·σ , mean uses σ/√n — not swapped?
Solve-back n uses ceil ? two-sided uses α/2 ?
Proportion normality via n·p̂(1−p̂)>5 , not n>30 ?
Answer ≥3 dp ? rounded when integer required ?