Formula Sheet for Exam P PDF

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Mark’s Formula Sheet for Exam P Discrete distributions • Uniform, U (m) – PMF: f (x) = – µ=

1 m,

for x = 1, 2, . . . , m

m2 − 1 m+1 and σ 2 = 2 12

• Hypergeometric – PMF: f (x) =

N1  N2  x

n−x

N  n

– x is the number of items from the sample of n items that are from group/type 1. N1 N2 N − n N1 ) – µ = n( ) and σ 2 = n( )( )( N N N N −1 • Binomial, b(n, p)   n x p (1 − p)n−x , for x = 0, 1, . . . , n – PMF: f (x) = x – x is the number of successes in n trials. – µ = np and σ 2 = np(1 − p) = npq

– MGF: M (t) = [(1 − p) + pet ]n = (q + pet )n • Negative Binomial, nb(r, p)   x−1 r p (1 − p)x−r , for x = r, r + 1, r + 2, . . . – PMF: f (x) = r−1 – x is the number of trials necessary to see r successes. 1 r r(1 − p) rq – µ = r( ) = and σ 2 = = 2 p p p2 p r  (pet )r pet – MGF: M (t) = = [1 − (1 − p)et ]r 1 − qet • Geometric, geo(p) – PMF: f (x) = (1 − p)x−1 p, for x = 1, 2, . . .

– x is the number of trials necessary to see 1 success. – CDF: P (X ≤ k) = 1 − (1 − p)k = 1 − q k and P (X > k) = (1 − p)k = q k 1−p q 1 – µ = and σ 2 = 2 = 2 p p p pet pet – MGF: M (t) = = 1 − (1 − p)et 1 − qet – Distribution is said to be “memoryless”, because P (X > k + j|X > k) = P (X > j).

• Poisson

λx e−λ , for x = 0, 1, 2, . . . x! – x is the number of changes in a unit of time or length. – PMF: f (x) =

– λ is the average number of changes in a unit of time or length in a Poisson process. – CDF: P (X ≤ x) = e−λ (1 + λ + – µ = σ2 = λ

– MGF: M (t) = eλ(e

λ2 2!

+··· +

λx x! )

t −1)

Continuous Distributions • Uniform, U (a, b) 1 , for a ≤ x ≤ b b−a x−a , for a ≤ x ≤ b – CDF: P (X ≤ x) = b−a a+b (b − a)2 – µ= and σ 2 = 12 2 etb − eta – MGF: M (t) = , for t 6= 0, and M (0) = 1 t(b − a)

– PDF: f (x) =

• Exponential 1 −x/θ e , for x ≥ 0 θ – x is the waiting time we are experiencing to see one change occur.

– PDF: f (x) =

– θ is the average waiting time between changes in a Poisson process. (Sometimes called the “hazard rate”.) – CDF: P (X ≤ x) = 1 − e−x/θ , for x ≥ 0.

– µ = θ and σ 2 = θ 2 1 – MGF: M (t) = 1 − θt – Distribution is said to be “memoryless”, because P (X ≥ x1 + x2 |X ≥ x1 ) = P (X ≥ x2 ).

• Gamma

1 1 xα−1 e−x/θ , for x ≥ 0 xα−1 e−x/θ = (α − 1)!θ α Γ(α)θ α – x is the waiting time we are experiencing to see α changes. – PDF: f (x) =

– θ is the average waiting time between changes in a Poisson process and α is the number of changes that we are waiting to see. – µ = αθ and σ 2 = αθ 2 1 – MGF: M (t) = (1 − θt)α • Chi-square (Gamma with θ = 2 and α = – PDF: f (x) =

r 2)

1 xr/2−1 e−x/2 , for x ≥ 0 Γ(r/2)2r/2

– µ = r and σ 2 = 2r – MGF: M (t) =

1 (1 − 2t)r/2

• Normal, N (µ, σ 2 ) – PDF: f (x) =

1 2 2 √ e−(x−µ) /2σ σ 2π

– MGF: M (t) = eµt+σ

2 t2 /2

Integration formulas Z 1 1 ′′ 1 ax • p(x)eax dx = p(x)eax − 2 p′ (x)eax + 3 p (x)e − . . . a a a  Z ∞  1 −x/θ e dx = (a + θ)e−a/θ • x θ a  Z ∞  1 −x/θ e dx = ((a + θ)2 + θ 2 )e−a/θ • x2 θ a Other Useful Facts • σ 2 = E[(X − µ)2 ] = E[X 2 ] − µ2 = M ′′(0) − M ′ (0)2 • Cov(X, Y ) = E [(X − µx )(Y − µy )] = E[XY ] − µx µy • Cov(X, Y ) = σxy = ρσx σy and σxy ρ= σx σy • Least squares regression line: y = µy + ρ

σy (x − µx ) σx

• When variables X1 , X2 , . . . , Xn are not pairwise independent, then n n X X X σi2 + 2 Var( Xi ) = σij i=1

and Var(

n X i=1

i=1

ai Xi ) =

n X i=1

i...