Stat 410: Random Variables and Confidence Intervals - Review of Dr. D. Scott's Lecture, Study notes of Statistics

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Stat 410 Random Variable

and Confidence Interval

Review

Dr. D. Scott

August 23, 2005

Fundamental object is the density function:

X ∼ f (x) = f (x 1

, x 2

,... , x p

which encodes ”structure.”

Sometimes one of the variables is labeled dif-

ferently:

(X, Y ) ∼ f (x, y) = f (x 1

,... , x p

, y).

Here, Y is the dependent or response vari-

able, while X 1

,... , X

p

are the independent

or predictor variables.

Properties of estimator

μˆ = ¯X =

n

n

∑

i=

X

i

E[ˆμ] = E





n

n

∑

i=

X

i





n

n

∑

i=

E[X

i

] =

nμ x

n

= μ x

so unbiased.

Also

X → μ x

(consistency) since

var( ¯X) = σ

2

¯ X

n

σ

2

X

→ 0 as n → ∞

by Chebyshev’s inequality.

σ

2

¯x

= E( ¯X − μ ¯x

2

= E





n

n

∑

i=

(X

i

− μ)





2

n

2

E





n

∑

i=

(X

i

− μ)

2

∑

i 6 =j

(X

i

− μ)(X j

− μ)





n

2

· n σ

2

x

n

2

n(n − 1) · 0 · 0 Why?

σ

2

x

n

CLT:

∑

of i.i.d. r.v.’s ≈ Normal. (RVLS)

Confidence Intervals for Parameters: pivots

Rearrange

P rob(− 1. 96 <

X − μ

σ/

n

to get

P rob( ¯X − 1. 96

σ

n

< μ <

X + 1. 96

σ

n

(± 2 .576 for a 99% confidence interval)

Pivot for σ

2

: S

2

1

n− 1

∑

(X

i

X)

2

∑

(X

i

X)

2

∑ [

(X

i

− μ) − ( ¯X − μ)

]

2

∑

(X

i

− μ)

2

− 2( ¯X − μ)

∑

(X

i

− μ) + n( ¯X − μ)

2

Now,

∑

(X

i

− μ) = n( ¯X − μ), so

∑

(X

i

X)

2

∑

(X

i

− μ)

2

− n( ¯X − μ)

2

or, dividing by σ

2

and rearranging,

∑

(

X

i

X

σ

)

2

(

X − μ

σ/

n

)

2

∑

(

X

i

− μ

σ

) 2

Since E[χ

2

(p)] = p and V ar[χ

2

(p)] = 2p,

E

[

(n − 1)

σ

2

S

2

]

= n − 1

or

E[S

2

] = σ

2

(unbiased)

Now have a pivot for a C.I. for σ

2

P rob

(

a <

(n − 1)S

2

σ

2

< b

)

= 1 − α

iff

P rob

(

(n − 1)S

2

b

< σ

2

(n − 1)S

2

a

)

where P r(χ

2

n− 1

< a) = P r(χ

2

n− 1

> b) =

α

(note: show R code to plot χ

2

A century ago, ”Student” (Gossett) showed

X − μ

σ( ¯X)

∼ N (0, 1)

and

X − μ

S( ¯X)

∼ T

n− 1

called Student’s t-distribution. Can show

T

n− 1

Z

√

χ

2

n− 1

/(n − 1)

Thus a C.I. for μ follows:

P rob(−a < T n− 1

< a) = 1 − α

becomes

P rob(μ ∈

X ± a · S( ¯X)) = 1 − α

which should be compared to

P rob(μ ∈

X ± 1. 96 σ( ¯X)) = 95%

Finally, the F distribution is a pivot for com-

paring variances.

F =

χ

2

(ν 1

)/ν 1

χ

2

(ν 2

)/ν 2

∼ F (ν 1

, ν 2

) = F

ν 1

,ν 2

Recall: T n− 1

Z

√

χ

2

n− 1

/(n − 1)

Thus

T

2

n− 1

Z

2

χ

2

n− 1

/(n − 1)

χ

2

χ

2

n− 1

/(n − 1)

= F

1 ,n− 1