1 Computational (or short-cut formula) for the variance

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1 Computational (or short-cut formula) for the variance

Transcript Of 1 Computational (or short-cut formula) for the variance

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Computational (or short-cut formula) for the variance, covariance, and correlation

σ2 = E(X2) − [E(X)]2,

Cov(X, Y ) = E(XY ) − E(X)E(Y ),

Cov(X, Y )

ρ(X, Y ) =

.

σX σY

TABLE OF COMMON DISTRIBUTIONS

Binomial(n, p) pmf mean and variance Hypergeometric(n,M,N)
pmf
mean and variance
Negative binomial(r,p) pmf
mean and variance Poisson(λ)
pmf mean and variance Discrete uniform pmf mean and variance

p(x) = nx px(1 − p)n−x, x = 0, 1, . . . , n E(X) = np, Var(X) = np(1 − p)

M N −M

p(x) = x Nn−x , max(0, n − N + M ) ≤ x ≤ min(n, M )

n

M

N −n M

M

E(X) = n , Var(X) =

n 1−

N

N −1 N

N

p(x) = x+r−r−1 1 pr(1 − p)x, x = 0, 1, 2, . . .

r(1 − p)

r(1 − p)

E(X) = p , Var(X) = p2

e−λλx

p(x) =

, x = 0, 1, 2, . . .

x!

E(X) = λ, Var(X) = λ

p(x) = 1/k, x = 1, 2, . . . , k

k+1

(k + 1)(k − 1)

E(X) =

, Var(X) =

2

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Normal(µ, σ2)
pdf
mean and variance Gamma(α, β)
pdf
mean and variance Exponential(λ) pdf mean and variance Chi Squared(n)
pdf
mean and variance Uniform(a, b)
pdf mean and variance

f (x) = √ 1 e−(x−µ)2/2σ2 , 2πσ
E(X) = µ, Var(X) = σ2

x∈R

f (x) = 1 xα−1e−x/β, βαΓ(α)

x≥0

E(X) = αβ, Var(X) = αβ2

f (x) = λe−λx, x ≥ 0 E(X) = 1/λ, Var(X) = 1/λ2

f (x) =

1

x(n/2)−1e−x/2,

2n/2Γ(n/2)

E(X) = n, Var(X) = 2n

x≥0

1

f (x) =

, a≤x≤b

b−a

E(X) = (a + b)/2, Var(X) = (b − a)2/12

Rule of thumb If sample size n > 30, then the Central Limit Theorem (CLT) can be applied.

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DESCRIPTIVE STATISTICS

1n X = n Xi,
i=1

S2 = 1

n

1

(Xi − X)2 =

n−1

n−1

i=1

n

(

Xi2 −

i=1

n i=1

Xi)2

n

TEST STATISTICS

Z = √n X − µ0 , σ

T = √n X − µ0 , S

√ Z= n

pˆ − p0 p0(1 − p0)

CI’s for µ (σ is known) CI’s for µ (σ is unknown) CI for σ2

CONFIDENCE INTERVALS

σ

σ

X − zα/2 √ , X + zα/2 √

n

n

σ −∞, X + zα √

σ and X − zα √ , +∞

n

n

S

S

X − tα/2,n−1 √ , X + tα/2,n−1 √

n

n

S −∞, X + tα,n−1 √

S and X − tα,n−1 √ , +∞

n

n

(n − 1)S2 (n − 1)S2

χ2

, χ2

α/2,n−1

1−α/2,n−1

(n − 1)S2

(n − 1)S2

0, χ21−α,n−1 and

χ2α,n−1 , +∞

Large-sample CI for p

pˆ(1 − pˆ)

pˆ(1 − pˆ)

pˆ − zα/2 n , pˆ + zα/2 n

Large-sample CI for arbitrary θ

θˆ − zα/2Sθˆ, θˆ + zα/2Sθˆ where Sθˆ is an estimator of the standard deviation σθˆ of θˆ
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