Distributions Derived the Normal Distribution

Distributions Derived from Normal Random Variables

Distributions Derived From the Normal Distribution

MIT 18.443

Dr. Kempthorne

Spring 2015

MIT 18.443

Distributions Derived From the Normal Distribution

1

Distributions Derived from Normal Random Variables

Outline

2, t, and F Distributions Statistics from Normal Samples

1 Distributions Derived from Normal Random Variables 2, t, and F Distributions

Statistics from Normal Samples

MIT 18.443

Distributions Derived From the Normal Distribution

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Distributions Derived from Normal Random Variables

Normal Distribution

2, t, and F Distributions Statistics from Normal Samples

Definition. A Normal / Gaussian random variable X N(?, 2)

has density function:

f (x ) = 1 e-(x-?)2/22 , - < x < +.

2

with mean and variance parameters:

? = E [X ]

=

J +

-

xf

(x

)dx

2 = E [(X - ?)2] = J-+(x - ?)2f (x)dx

Note: - < ? < +, and 2 > 0.

Properties:

Density function is symmetric about x = ?.

f (? + x) = f (? - x).

f (x) is a maximum at x = ?.

f ""(x) = 0 at x = ? + and x = ? -

(inflection points of bell curve)

Moment generating function:

MX (t) = E [etX ] = e?t+2t2/2

MIT 18.443

Distributions Derived From the Normal Distribution

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Distributions Derived from Normal Random Variables

Chi-Square Distributions

2, t, and F Distributions Statistics from Normal Samples

Definition. If Z N(0, 1) (Standard Normal r.v.) then U = Z 2 12,

has a Chi-Squared distribution with 1 degree of freedom.

Properties:

The density function of U is:

fU (u)

=

u-1/2 e-u/2, 0 < u <

2

Recall the density of a Gamma(, ) distribution:

g (x)

=

()

x

-1e

-x

,

x

>

0,

So U is Gamma(, ) with = 1/2 and = 1/2.

Moment generating function MU (t) = E [etU ] = [1 - t/]- = (1 - 2t)-1/2

MIT 18.443

Distributions Derived From the Normal Distribution

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Distributions Derived from Normal Random Variables

Chi-Square Distributions

2, t, and F Distributions Statistics from Normal Samples

Definition. If Z1, Z2, . . . , Zn are i.i.d. N(0, 1) random variables V = Z12 + Z22 + . . . Zn2

has a Chi-Squared distribution with n degrees of freedom.

Properties (continued)

The Chi-Square r.v. V can be expressed as:

V = U1 + U2 + ? ? ? + Un

where U1, . . . , Un are i.i.d 21 r.v.

Moment generating function

MV (t) = E [etV ] = E [e

t(U1+U2+???+Un)] = E [etU1 ]

? ? ? E [etUn ] = (1 - 2t)-n/2

Because Ui are i.i.d. Gamma( = 1/2, = 1/2) r.v.,s

V Gamma( = n/2, = 1/2).

Density

function:

f (v )

=

1 2n/2(n/2)

v

(n/2)-1

e

-v

/2

,

v

>

0.

( is the shape parameter and is the scale parameter)

MIT 18.443

Distributions Derived From the Normal Distribution

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Distributions Derived from Normal Random Variables

Student's t Distribution

2, t, and F Distributions Statistics from Normal Samples

Definition. For independent r.v.'s Z and U where

Z N(0, 1) U 2r the distribution of T = Z / U/r is the

t distribution with r degrees of freedom.

Properties

The density function of T is

[(r + 1)/2]

t2 -(r +1)/2

f (t) =

1+

, - < t < +

r (r /2)

r

For what powers k does E [T k ] converge/diverge?

Does the moment generating function for T exist?

MIT 18.443

Distributions Derived From the Normal Distribution

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Distributions Derived from Normal Random Variables

F Distribution

2, t, and F Distributions Statistics from Normal Samples

Definition. For independent r.v.'s U and V where

U 2m

V 2n

U /m

the distribution of F =

is the

V /n

F distribution with m and n degrees of freedom.

(notation F Fm,n) Properties

The density function of F is

[(m + n)/2] f (w ) =

m n/2 w m/2-1

(m/2)(n/2) n

with domain w > 0.

m 1+ w

n

-(m+n)/2

,

E [F ]

=

E [U/m] ? E [n/V ]

=

1 ? n ?

1 n-2

=

n n-2

(for

n

>

2).

If T tr , then T 2 F1,r .

MIT 18.443

Distributions Derived From the Normal Distribution

7

Distributions Derived from Normal Random Variables

Outline

2, t, and F Distributions Statistics from Normal Samples

1 Distributions Derived from Normal Random Variables 2, t, and F Distributions

Statistics from Normal Samples

MIT 18.443

Distributions Derived From the Normal Distribution

8

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