MAT2377 Cheat Sheet (DRAFT) by t847222

P(A) = N(A)/N(S)
P(A) = f(A)/n

P(A|B) = P(A∩B)­/P(B)

F(x) = P(X≤x) = Σf(x_i)

p(x,y) = P(X=x, Y=y) = P({X=x­}∩{­Y=y})

X = # of trials until 1^st success
X ~ g(p)
f(x) = (1-p)^x-1p, for x=1,2,...
F(x) = 1-(1-p)^x, for x=1,2,...
E[X] = 1/p
V[X] = (1-p)/p²

P(a<X<b) = ∫f(x)dx = F(b)-F(a)
f(x) = F'(x)
F(x) = P(X<x) = ∫f(t)dt
E[X] = ∫xf(x)dx
V[X] = ∫x²f(x)dx­-E[X]²
E[g(X)] = ∫g(x)f­(x)dx
V[g(X)] = ∫(g(x))²f(x)dx­=E[­g(X)]²

f(x) = 1/√(2πσ²)*e^{-(x-μ)­^2/­(2σ^2)}, -∞<­x<∞
X ~ N(μ, σ²)
E[X] = μ
V[X] = σ²

x̄ = Σx_i/n

P = Y/n
Y ~ b(n,p)
Z = (P-p)/­√(p­(1-p)n) ~ N(0,1)
E = z_[α/2­]√(­p(1­-p)/n)

r = cov/(s­_xs_y)

n! = n(n-1)­(n-­2)*...*1 if n≥1
 ­ ­ ­ = 1                         if n=0
nPr = n!/(n-r)!

f(x) = P(X=x)

σ² = V[X] = Σx²f(x)-E[X]²

E[g(X,Y)] = Σ^xΣ^yg(x,y)­p(x,y)
E[X] = Σ^xxp(x)
E[Y] = Σ^yyp(y)
E[X+Y] = E[X]+E[Y]
Cov[X,Y] = (Σ^xΣ^yxyp(x,­y))­-E[­X]E[Y]
V[X+Y] = V[X]+V­[Y]­+2C­ov[X,Y]

X = # of event in time [0,1]
p(x) = e^-μ*μ^x/x!, for x=0,1,...
X ~ P(μ)
E[X] = V[X] = μ
Approx­ima­tion: binomial f(x) ≈ p(x), μ=np
Process: between [0,t], μ=λt

f(x) = 1/(b-a), a≤x≤b
 ­ ­ ­ ­ ­ = 0,  ­ ­ ­ ­ ­ ­ ­ ­ ­els­ewhere
X ~ U[a,b]
E[X] = (a+b)/2
V[X] = (b-a)²/12

s² = ((Σx²_i)-nx̄²)/(n-1)

Z = (X̄-μ)­/(σ/√n)
X̄ ~ N(μ, σ²/n) ⇒ Z ~ N(0,1)

α = P(Z>z_α) = 1-Φ(z)
μ ∈ [x̄-E, x̄+E]
σ² known: E = z_[α/2­]*σ/√n
σ² unknown: T = (X̄-μ)­/(S/√n) ~ T(n-1)
 ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­P(T­>t_­[α,v]) = α; z_α = t_[α,∞]
 ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ E = t_[α/2­,n-­1]*s/√n
σ² unknown, n≥40: (X̄-μ)­/(S/√n) ~ N(0,1)
 ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ E = z_[α/2­]*s/√n
n≥((z_­[α/­2]σ)/E)²

n = n_1*...*n_k
nCr = (ⁿr) = n!/r!(­n-r)!

P(A∩B) = P(B|A)P(A) = P(A|B)P(B)
 ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ = P(A)P(B) if ind.

E[g(X)] = Σg(x)f(x)
V[g(X)] = [Σ(g(x))²f(x)]-­(E[­g(X)])²

S = {success, failure} = {p,q}
p = P(I=1)
I ~ Ber(p)
E[I] = p
V[I] = p(1-p)

X = # of trials to until r^th success
X ~ Nb(r,p)
f(x) = (^x-1r-1)(1-p)^x-rp^r, for x=r,r+­1,...
E[X] = r/p
V[X] = r(1-p)/p²

T = time until r^th outcome of Poisson process
F(x) = P(T≤x) = 1-P(T>x)
 ­ ­ ­ ­ ­ ­ = 1-Σ^r-1e^-λx(λx)^k/k!
E[T] = r/λ
V[T] = r(1-λ)/λ²

Z = (X-E[X­])/­√(V[X])
F(x) = P(X≤x) = Ф((x-μ)/σ)
P(a<X<b) = F(b)-F(a)

Rank of k^th percen­tile: (n+1)*­k/100 = m+p, 0≤p<1
k^th percentile = y_m+p(­y_[­m+1­]-y_m)
IQR = q_3-q_1

Null hyp: make no change
Alternate hyp: test according to question
⇒Test 1: μ ≠ μ_0; 2: μ > μ_0; 3: μ < μ_0;
Confidence interval decision: reject H_0 for H_1 if μ_0 is not in confidence interval
Z_0 or T_0 decision:
σ² known: Z_0 = (X̄-μ_­0)/­(σ/√n) ~ N(0,1)
Test 1: reject if |z_0| > z_[α/2]; 2: z_0 > z_α; 3: z_0 < -z_α
σ² unknown: T_0 = (X̄-μ_­0)/­(S/√n) ~ T_[n-1]
Test 1: |t_0| > t_[α/2­,n-1]; 2: t_0 > t_[α,n-1]; 3: t_0 < -t_[α,n-1]
Pop. & σ² unknown: replace σ with S from σ² known
p-Value decision: reject if p-value < α
p-value = 2[1-Ф(­|z_­0|)], test 1 & z-value
= 1-Ф(z_0), test 2 & z-value
= Ф(z_0), test 3 & z-value
= 2P(T>|­t_0|), test 1 & t-value
= P(T>t_0), test 2 & t-value
= P(T<t_0), test 3 & t-value

P(A∩B') = P(A)-P­(A∩B)
P(A∪B) = P(A)+P­(B)­-P(A∩B)
P(A'∩B') = 1-P(A∪B)
P(A∪B∪C) = P(A)+P­(B)­+P(­C)-­P(A­∩B)­-P(­A∩C­)-P­(B∩­C)+­P(A­∩B∩C)
P(A_1∪...∪­A_n) = 1-P(A_­1'∩...∩­A_n')

μ = E[X] = Σxf(x)

p(x) = P(X=x) = Σ^yp(x,y)
p(y) = P(Y=y) = Σ^xp(x,y)

X = # of successes from n trials
X ~ b(n,p)
f(x) = (ⁿx)p^x(1-p)^n-x, for x=0,1,...,n
E[X] = np
V[X] = np(1-p)

Waiting time
X ~ Exp(λ)
f(x) = λe^-λx, x>0
F(x) = 1-e^-λx, x>0
E[X] = 1/λ
V[X] = 1/λ²
Lack of memory: P(X>s+­t|X­>s) = P(X>t)

Z ~ N(0,1)
PMF: ⌀(z) = 1/√(2π)*e^{-1/2*z^2}
CDF: Φ(z) = P(Z≤z) = ∫⌀(t)dt
Φ(0) = 0.5
P(Z≤-z) = P(Z≥z)
Φ(-z) = 1-Φ(z)
P(a≤Z≤b) = Φ(b)-Φ(a)
P(-a≤Z≤-b) = Φ(a)-Φ(b)

Y ~ N(μ_Y, σ²_Y)
E[Y] = Σc_iE[X_i]
V[Y] = Σc²_iV[X_i]²
X̄ = 1/nΣX_i
E[X̄] = μ
V[X̄] = σ²/n

cov = ((Σx_i­y_i­)-(­Σx_­i)(­Σy_­i)/­n)/­(n-1)

y = a+Bx
B = ((Σx_i­y_i­)-(­Σx_­i)(­Σy_­i)/­n)/((Σx²_i)-(Σx_i)²/n)