Statistical Formulas Cheat Sheet for Final Exam, Cheat Sheet of Statistics

Download Statistical Formulas Cheat Sheet for Final Exam and more Cheat Sheet Statistics in PDF only on Docsity! Formula Sheet for Final Exam Summary Statistics • Sample mean: x̄ = n∑ i=1 xi n • Sample variance: s2 = ∑n i=1(xi − x̄)2 n− 1 • Sample standard deviation:= √ s2 • Inter-quartile range = q75 − q25, where qx = xth percentile. Probability • Complement: P (Ac) = 1− P (A) • Addition law: P (A or B) = P (A) + P (B)− P (A and B) • Conditional probability: P (A|B) = P (A and B) P (B) • If A and B are mutually exclusive: P (A and B) = 0 • If A and B are independent: P (A and B) = P (A)P (B) • Bayes’ rule: P (A|B) = P (B|A)P (A) P (B) • Partition law: If A1, . . . , An are mutually exclu- sive and ∑n i=1 P (Ai) = 1, then P (B) = n∑ i=1 P (B|Ai)P (Ai) Discrete Distribution • E(X) = ∑ x∈X xP (x) • V (X) = ∑ x∈X (x− µ)2P (x) = ∑ x∈X x2P (x)− µ2 • If X ∼ Bernoulli (p), p(x) = { p if x=1 1− p if x=0 E(X) = p V (X) = p(1− p) • If X ∼ Binomial (n, p), for r = 0, 1, . . . n, P (X = r) = ( n r ) pr(1− p)n−r E(X) = np V (X) = np(1− p)( n r ) = n!r!∗(n−r)! where n! = n(n − 1)(n − 2) · · · 1 and 0! = 1 • If X ∼ Poisson ( λ ), for k = 0, 1, 2, . . . P (X = k) = λk k! e−λ E(X) = λ, V (X) = λ Continuous Distribution • Let F (xo) be the cumulative distribution func- tion of X with density p(x): F (xo) = P (X ≤ xo) = ∫ xo −∞ p(x)dx P (a < X ≤ b) = F (b)− F (a) • If X is Normal with mean µ and variance σ2: X ∼ N (µ, σ2), then X − µ σ = Z ∼ N (0, 1) Expectation and Variance Let X and Y denote two independent random vari- ables and let a and b denote two known constants. • E(aX + b) = aE(X) + b 1 • V (aX + b) = a2V (X) • V (X) = E(X2)− [E(X)]2 • E(X + Y ) = E(X) + E(Y ) • V (X + Y ) = V (X) + V (Y ) Sampling Distribution Let X1, . . . , Xn be an iid sample with E(Xi) = µ and V (Xi) = σ 2. Denote X̄ the sample mean and s2 the sample variance. X̄ ∼ N ( µ, σ2 n ) X̄ − µ σ/ √ n ∼ N(0, 1) X̄ − µ s/ √ n ∼ tn−1 Point Estimation Let X and Y be estimators for θ. • Bias = E(X)− θ • MSE (X) = E[(X − θ)2] = V(X) + [Bias(X)]2 • Efficiency of X compared to Y = MSE(Y ) MSE(X) *MSE = mean squared error Confidence Interval for One Mean • If the population variance σ2 is known: x̄± z(1−C%)/2 × σ√ n . where z(1−C%)/2 is the (1 − C%)/2 quantile of the standard Normal distribution. • If the population variance σ2 is unknown: x̄± t(1−C%)/2, n−1 × s√ n . Confidence Interval for the Difference in Two Means • If σ21 and σ22 are known: (x1 − x2)± z(1−C%)/2 × √ σ21 n1 + σ22 n2 . • If σ21 and σ22 are unknown and equal: (x1 − x2)± t(1−C%)/2, n1+n2−2 × sp √ 1 n1 + 1 n2 . sp = √ (n1 − 1)s21 + (n2 − 1)s22 n1 + n2 − 2 . • If σ21 and σ22 are unknown and unequal: (x1 − x2)± t(1−C%)/2, tWS × √ s21 n1 + s22 n2 tWS = (s21/n1 + s 2 2/n2) 2 (s21/n1) 2/(n1 − 1) + (s22/n2)2/(n2 − 1) . Hypothesis Testing • Type I error (α) = P (reject H0|H0 is true) • Type II error (β) = P (not reject H0|H0 is false) • Power = 1− β • Test statistic for one mean, H0 : µ = µ0:, X̄ − µ0 σX̄ • Test statistic for difference of two sample means, H0 : µ1 − µ2 = d0, (X1 −X2)− d0 σX1−X2 Joint Distribution Let X and Y denote two random variables with joint distribution p(x, y). Let a and b denote two known constants • E[g(X,Y )] = ∑ xy g(x, y)p(x, y) • V [g(X,Y )] = ∑ xy (g(x, y)− E[g(x, y)]) 2 p(x, y) 2