Cheatography

# MAT2377 Cheat Sheet (DRAFT) by t847222

Intro to Probability and Statistics final exam cheat sheet

This is a draft cheat sheet. It is a work in progress and is not finished yet.

### Classical + Relative

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

### Condit­ional

 P(A|B) = P(A∩B)­/P(B)
A given B

### CDF

 F(x) = P(X≤x) = Σf(x_i)

### Joint PMF

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

### Geometric Distri­bution

 X = # of trials until 1st 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

### Normal Distri­bution

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

### Sample Mean

 x̄ = Σx_i/n

### Box Plot

Describe histogram: skewness, uni/bi­modal

### Constr­ucting Confidence Interval

 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)

### Sample Correl­ation

 r = cov/(s­_xs_y)
s_x and s_y are standard dev.

### Permut­ations

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

### PMF

 f(x) = P(X=x)

### Variance

 σ2 = V[X] = Σx2f(x)-E[X]2
Standard deviation = sqrt(V[X])

### Joint Properties

 E[g(X,Y)] = ΣxΣyg(x,y)­p(x,y) E[X] = Σxxp(x) E[Y] = Σyyp(y) E[X+Y] = E[X]+E[Y] Cov[X,Y] = (ΣxΣyxyp(x,­y))­-E[­X]E[Y] V[X+Y] = V[X]+V­[Y]­+2C­ov[X,Y]

### Poisson Distri­bution

 X = # of event in time [0,1] p(x) = e-μ*μx/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

### Continuous Uniform Distri­bution

 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

### Sample Variance

 s2 = ((Σx2_i)-nx̄2)/(n-1)

### CLT

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

### Confidence Level

 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

### Combin­ations

 n = n_1*...*n_k nCr = (nr) = n!/r!(­n-r)!
Order doesn't matter

### Multip­luc­ation Rule

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

### Transf­orm­ation

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

### Bernoulli Trial

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

### Negative Binomial Distri­bution

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

### Erlang Distri­bution

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

### Standa­rdi­zation Thm

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

### Percentile

 Rank of kth percen­tile: (n+1)*­k/100 = m+p, 0≤p<1 kth percentile = y_m+p(­y_[­m+1­]-y_m) IQR = q_3-q_1
Median is 50th percentile

### Hypothesis

 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

 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')

### Expected Value

 μ = E[X] = Σxf(x)

### Marginal PMF

 p(x) = P(X=x) = Σyp(x,y) p(y) = P(Y=y) = Σxp(x,y)

### Binomial Distri­bution

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

### Expone­ntial Distri­bution

 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)

### Standard Normal Distri­bution

 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)

### Linear Combin­ation

 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
Y = c_1X_1­+...+c­_nX_n

### Sample Covariance

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

### Line of Best Fit

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