Summary 394-5 - Contains list of formulas from MATH 394,395 that you will need for MATH 396 PDF

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Summary

Contains list of formulas from MATH 394,395 that you will need for MATH 396...

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MATH/STAT 395A

Final Exam: Formula Sheet

Winter 2018, Wed March 14

1. Permutations and combinations Qn There are n! = i=1 i = 1.2.3.4. . . . n permutations of n objects. n There are ( ) = n!/(k!(n − k)!) ways of choosing a given k objects from n. k 2. Joint and conditional probabilities S T If C and D are any events: P(C D) = P(C) + P(D) − P(C D).

The conditional probability of C given D is P(C | D) = P(C T C and D are independent if P(C D) = P(C).P(D).

T

D) / P(D).

3. Laws and theorems T S S S Suppose E1 , . . . , Ek is a partition of Ω. That is Ei Ej is empty, and E1 E2 . . . Ek = Ω. Pk T Pk The law of total probability states that: P(D) = j=1 P(D | Ej ) P(Ej ) j=1 P(D Ej ) = Bayes’ Theorem states that: P(Ei | D) = P(D | Ei ) P(Ei )/P(D) 4. Random variables and distributions Probability mass/density function Cumulative dist. func. CDF, P(X ≤ x) Joint mass/density func. of (X, Y ) Marginal mass/density of X

Conditional of X given Y = y Independence of X and Y

discrete (mass) pmf: P(X = x) = pX (x) FX (x) =

w≤x p X (w)

P

pX,Y (x, y) = P(X = x, Y = y) P pX (x) = y pX,Y (x, y)

pX|Y (x|y) = pX,Y (x, y)/pY (y) pX,Y (x, y) = pX (x).pY (y)

or All assumed to hold for

pX|Y (x|y) = pX (x) all X ∈ X , y ∈ Y

continuous (density) pdf: fX (x) R FX (x) = x−∞ fX (w)dw fX,Y (x, y ) R∞ y=−∞ fX,Y (x, y)dy fX|Y (x|y) = fX,Y (x, y)/fY (y) fX,Y (x, y) = fX (x)fY (y) fX (x) =

fX|Y (x|y) = fX (x) −∞ < x < ∞, − ∞ < y < ∞

5. Moments of random variables: (provided the relevant sums/integrals converge absolutely.) Expectation:

E(X) E(g(X))

Conditional expectation E(g(X)|Y = y)

xx

P(X = x) P x g(x) P(X = x) P

x g(x)

P

P(X = x | Y = y)

R∞

xfX (x)dx −∞ g(x)fX (x)dx R∞ −∞ g(x)fX|Y (x|y)dx R ∞−∞

(i) For any random variables X : Variance: var(X) = E((X − E(X))2 ) = E(X 2 ) − (E(X ))2 Note: E(aX + b) = aE(X) + b, var(aX + b) = a2 var(X ). (ii) For any random variables X, Y , Z and W : Covariance: cov(X, Y ) = E((X − E(X))(Y − E(Y ))) = E(XY ) − E(X )E(Y ) p Correlation: ρ(X, Y ) = cov(X, Y )/ var(X )var(Y ), − 1 ≤ ρ(X, Y ) ≤ 1 Note: E(X + Y ) = E(X) + E(Y ), var(X + Y ) = var(X) + var(Y ) + 2cov(X, Y ) cov(aX +b, cW +d) = ac cov(X, W ), cov(X +Y, W +Z) = cov(X, W )+cov(X, Z )+cov(Y, W )+cov(Y, Z ) (iii) Conditional expectation and variance Expectation of h(Y ) = E(X | Y ): E(h(Y )) = E(E(X|Y )) = E(X). Or more generally for E(g(X) | Y ): E(E(g(X)|Y )) = E(g(X). Variance of X: var(X) = E(var(X|Y )) + var(E(X|Y )) 6. A note about Normal (Gaussian) random variables (a) Linear transformations of Normal random variables are Normal (b) Linear combinations of independent Normal r.vs are Normal (c) Different linear combinations of independent Normal r.vs are called jointly Normal (d) If X and Y are jointly Normal and cov(X, Y )=0, then X and Y are independent.

pmf or pdf

7. Standard distributions: (a) Binomial; B(n, p) index n, parameter p (b) Geometric; Geom(p); parameter p (c) Neg. Binomial; N egB(r, p); index r, parameter p (d) Poisson: Po(µ) (e) Uniform on (a, b); U (a, b); (f) Normal; N (µ, σ 2 ) (g) Exponential; E(λ) rate parameter λ (h) Gamma, G(α, λ) shape α, rate λ

P(X = k) =

(

n k

)pk (1 − p)n−k

k = 0, 1, 2, ..., n P(X = k) = p(1 − p)k−1 k = 1, 2, 3, 4, ......

mean

variance

np

np(1 − p)

1/p

(1 − p)/p2

k−1

)pr (1 − p)k−r r/p r−1 k = r, r + 1, r + 2, .... P(X = k) = exp(−µ)µk /k!, k=0,1,2,... µ fX (x) = 1/(b − a), a < x < b (b + a)/2 √ 2 2 2 fX (x) = (1/ 2πσ ) exp(−(x − µ) /2σ ) µ P(X = k) =

(

fX (x) = λ exp(−λx) 0≤x...