AMS 311 Formulas for Test 2 PDF

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AMS 311 Formulas for Test 2 Some Possibly Useful Formulas for Exam Binomial(n,p): p(i) =

( ni ) p (1− p) i

n−i

, for i = 0,1,...,n

Mean is np; variance is np(1 − p);

mgf is M(t) = (pet + 1 − p)n Geometric(p): p(i) = (1 − p)i−1p, for i = 1,2,.... Mean is 1/p; variance is (1 − p)/p 2; mgf is M(t) = (pet)/[1 − (1 − p)et]

Hypergeometric: f(x) =

(ax)( Nn−−ax ) ( Nn )

, Mean is

na N

i Poisson(λ): p(i) = e− λ λ for i = 0, 1, 2,... , Mean and variance equal λ; i! mgf is M(t) = exp{λ(et − 1)}

1 , Mean is (a + b)/2, variance is (b−a) 2/12 b− a mgf is M(t) = [etb − eta]/[t(b − a)]. Uniform( a, b): f(x) =

1 e−( x−μ ) / 2 σ − ∞ < x < ∞; (X−µ)/ σ is Normal(0,1); √2 π σ mgf is M(t) = exp{µt + (σ 2t2)/2} 2

2

Normal(µ, σ2): f(x) =

Exponential(λ): F(a) = 1−e−λa, a ≥ 0; f(x) = λe−λx if x > 0. Mean is 1/λ, variance is 1/λ2. mgf is M(t) = λ/(λ − t) Gamma(n, λ): f(x) = λe−λx(λx)n−1 /(n−1)! , if x ≥ 0. Mean is n/λ, variance is n/λ2. mgf is M(t) = [λ/(λ − t)]n. Gamma Alternative pdf: If X Gamma(α , β) , then the pdf for X is given by

{

1 x α−1 e−x / β , x≥ 0 , α>0, β >0 f ( x )= Γ ( α ) β α 0, otherwise Cov(X,Y ) = E(XY )  E(X)E(Y); Var(X) = E(X2)  [E(X)]2

E ( X ) =αβ ,Var ( X )=α β

2...