Probability and Statistics Exam 1: Equations and Functions - Prof. Kobi Abayomi, Exams of Data Analysis & Statistical Methods

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ISYE 2028

Exam 1 Equations

Dr. Kobi Abayomi

March 25, 2009

You must show all work to receive full credit. All regrades must be submitted the day the

exam is returned.

Information You May Find Useful :

Functions of One Random Variable

Given a random variable X, defined with probability mass function pX :

F (x) =

all t≤x

p X

(t)

P(a ≤ X ≤ b) =

a≤t≤b

p X

(t)

μ =

all X

xp(x) =

all X

xP(X = x) = E(X)

V ar(X) = σ

2 = E[(X − μ)

2 ] = E(X

2 ) − [E(X)]

2

Given a random variable X, defined on the real line R, with probability density function f X

F

X

(x) =

x

−∞

f X

(t)dt

P(a ≤ X ≤ b) =

b

a

f (x)dx

μ X

R

xf (x)dx = E(X)

σ

2

X

= E[(X − μ)

2

] =

(x − μ)

2

f (x)dx = V ar(X) = V ar(X) = E(X

2

) − μ

2

Functions of Multiple Random Variables

For jointly continuous X, Y

F

X,Y

(x, y) = P(X ≤ x, Y ≤ y)

dF X,Y

(x, y) = f X,Y

(x, y)

p X

(x) = P(X = x) = C

n

x

p

x

(1 − p)

x

E(X) = np; V ar(X) = np(1 − p)

M GF = (1 − p + pe

t )

n

Poisson: P oi(λ)

pX (x) = P(X = x) =

e

−λ

λ

x

x!

E(X) = V ar(X) = λ

M GF = e

λ(e

t −1)

Canonical Probability Density Functions

Uniform: U (a, b):

f (x) =

1

b−a

, a < x < b

0 , o.w..

E(X) =

b + a

; V ar(X) =

(b − a)

2

M GF =

e

tb

− e

ta

t(b − a)

Normal: N (μ, σ

2

):

f (x) = (2πσ

2

)

− 1 / 2

exp{−

(x − μ)

2

2 σ

2

E(X) = μ, V ar(X) = σ

2

M GF = e

μt+σ

2 t

2 / 2

Gamma: Γ(α, β):

f (x; α, β) =

x

α− 1 e

−x/β

Γ(α)β

α

E(X) = αβ; V ar(X) = αβ

2

M GF = (1 − βt)

−α

Exponential: Exp(λ) ≡ Γ(α = 1, β =

1

λ

Chi-Squared: χ

2 (r) ≡ Γ(α =

r

2

, β = 2)

Test Statistics, Hypothesis Testing and Confidence Intervals

Let

θ be an estimate of a population parameter θ, often a sample mean

Let S.D.(

θ) =

V ar(θ)

n

; Let S.E.(

θ) =

S

2

n

If

θ ∼ N (E(

θ), V ar(

θ))

Then use Standard Normal Test Statistic

Z =

θ − θ

S.D.(

θ)

where

Z ∼ N (0, 1)

If

θ ∼ E(

θ) with V ar(

θ) unknown

Then use T-Distribution Test Statistic

T =

θ − θ

S.E.(

θ)

where

T ∼ t α,df =n− 1

Confidence interval for population variance

σ

2

∈ [

(n − 1)s

2

χ

2

α/ 2 ,n− 1

(n − 1)s

2

χ

2

1 −α/ 2 ,n− 1

]

Confidence interval for ratio of variances

σ

2

σ

1

s

2

2

F

1 −α/ 2 ,ν 1 ,ν 2

s

2

1

s

2

2

F

α/ 2 ,ν 1 ,ν 2

s

2

1