ISE261 Equations:

Binomial Distribution: b(x,n,p) = Cx,n px (1-p)n-x For 0,1,2,...,n E(X) = np

V(X) = np(1-p)

Negative Binomial: nb(x,r,p) = Cr-1,x+r-1 pr (1-p)x For 0,1,2,.. E(X) = r(1-p)/p

V(X) = r(1-p)/p2

Hypergeometric: h(x, n, M, N) = Cx,M * Cn-x,N-M / Cn,N E(X) = nM/N

V(X) = nM(N-M)(N-n)/N2(N-1)

Poisson: po(x, λ) = e–λ(λ)x/x!E(X) = λ

V(X) = λ+

Expected Value Discrete RV: E(X) = ∑ xi*p(xi) Continuous: E(X)=  x f(x)dx
-

Bayes' Theorem: P(Aj | B) = P(B|Aj)P(Aj) / ∑ P(B|Aj)P(Aj) For j = 1,...,k

Conditional Probability: P(A∩B) = P(A|B)*P(B)

Combination: Ck,n = n! / k!(n-k)! Permutation: Pk,n = n! / (n-k)!

Independence: P(A∩B) = P(A) * P(B)

Probability Property: P(A) = N(A) / N(T)

Addition Rule: P(A U B) = P(A) + P(B) - P(A∩B)

Complement: P(A) = 1 - P(A')

Uniform: f(x) = 1/ (b-a)for a ≤ x ≤ b

Z-transform: Z = (x - µ) / σ

Exponential CDF: F(x) = 1 – e–λt for t ≥ 0E(X) = 1/λ

V(X) = 1/λ2

Weibul CDF: F(x) = 1 – e–(x/) power α for x ≥ 0E(X) = β(1+1/α)

V(X)= 2{(1+2/)–[(1+1/)]2}

Gamma Function: (n) = (n-1)!( positive integer n); () = (-1) (-1); (1/2) = 

Lognormal CDF: F(x) = [(ln(x) - ln) / ln] where { is the CDF of Z} E(x)= e µ+(σσ)/2

-1 -1
Beta:f(x) = 1 (+) x-A B-xfor A  x  B
B-A ()() B-A B-A

E(X) = A + (B - A)( /( + ))

Gamma transform: F(x/, ) for x ≥ 0 E(x) =  V(x) = 2

ISE 261 Equations Continued:

Two sided-CI for Mean (known σ): (L,U) = x_bar +/- (z/2 σ/ √(n))

Two sided-CI for Mean (small n ≤ 40): (L,U) = x_bar +/- (t/2,v s / √(n))

Two sided-CI for Mean (large n; > 40): (L,U)= x_bar +/- (z/2 s / √(n))

Two sided-CI for Variance: (L) = (n-1)s2/ χ2/2, n-1 (U) = (n-1)s2/ χ21-/2, n-1

Sample Size for Full Bound W: n = (2 z/2 / W)2

Sample Size for Half Bound B: n = ( z/2 / B)2

Z-transform: Z = (x - µ) / σ

+ +
E[h(X,Y)]=h(x,y)f(x,y)dxdyFor Discrete X&Y: E[h(X,Y)]=h(x,y)p(x,y)
- - x y

Cov(X,Y) = E(XY) - uxuy

Corr(X,Y) = Cov(X,Y)For Sampling Distributions:
x y (X_bar) =  V(X_bar) = 2 / n
V(X) = E(X2) – E(X)2 E(T) = n V(T) = n2

HT for Mean (known σ)> Ζ =(x_bar - µ0) / (( / √(n))

HT for Mean (small n ≤ 40)> t = (x_bar - µ0) / ((s / √(n))

HT for Mean (large n; > 40)> Z = (x_bar - µ0) / ((s / √(n))

HT for Variance> χ2 = ((n-1) s2)/ 02

Beta Error for HT: a0 (Upper-Tail): β(u’) = [z+ ((u0 –u’)/ (/√(n))]

Beta Error for HT: a0 (Lower-Tail): β(u’) = 1 -[-z+ ((u0 –u’)/ (/√(n))]

Beta Error for HT: a≠0

β(u’) =[z/2+ ((u0 –u’)/ (/√(n))] - [-z/2+ ((u0 –u’)/ (/√(n))]

P_value for One Sided Tests: Upper tailed: P = 1 – Φ(z); Lower:P = Φ(z)
Two Sided Test: P_value = 2[1– Φ(|z|)]

Determining n ( known):
One-Sided HT for a given β(u’): n =  (z+ z)2
u0 - u
Two-Sided HT for a given β(u’): n =  (z/2+ z)2
u0 - u

Marginal pdf: +
fx(x) = f(x,y)dy for -x +
-

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