(15) chi-square, student’s t and snedecor’s f distributions

Applied Sta+s+cs and Compu+ng Lab

Chi-‐square, Student’s t and Snedecor’s F Distribu7ons

Applied Sta+s+cs and Compu+ng Lab Indian School of Business

Applied Sta+s+cs and Compu+ng Lab 2

Learning Goals

•  To become familiar with Chi-‐square, t and F •  To get to know the rela7onships among Normal, Chi-‐square, t and F

•  To get to know the uses of Chi-‐square, t and F


Chi-‐square distribu7on Rela7on to normal distribu7on

•  If X has a standard normal distribu7on, then its square has a Chi-‐square distribu7on with 1 degree of freedom.

•  In fact, sum of squares of n independent standard normal variables has a chi-‐square distribu7on with n degrees of freedom.

3


Chi-‐square distribu7on -‐ Uses

•  Very useful in – Hypothesis tes7ng – Construc7on of confidence intervals •  For the variance of a normal distribu7on based on a random sample

4


Chi-‐square Distribu7on •  The probability density func7on (pdf) of the chi-‐squared distribu7on

is

where Γ(k/2) denotes the Gamma func7on. Gamma func7on has closed-‐form values for integer k. where k is an integer.

⎪⎪

⎩

⎪⎪

⎨

⎧≥

Γ=

−−

Otherwise

xkex

kxf k

xk

,0

0,)2(2),( 2

21

2

)!1()( −=Γ kk

π=⎟⎠

⎞⎜⎝

⎛Γ21

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Density of Chi-‐square distribu7on with k degrees of freedom

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Chi-‐square distribu7on Proper7es

•  Let X have a chi-‐square distribu7on with k degrees of freedom (df). This is denoted by

•  Mean of X = k •  Variance of X = 2k •  If X and Y have independent Chi-‐square distribu7ons with

degrees of freedom m and n respec7vely, then X+Y has a chi-‐square distribu7on with degrees of freedom m+n.

2~ kX χ

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‘t distribu7on’ Rela7on to normal and Chi-‐square distribu7ons

•  Suppose has a standard normal distribu7on and has a chi-‐square distribu7on with k degrees of freedom . Suppose further that and are independent.

•  Define •  The distribu7on of is called a Student’s t distribu7on (or

simply t distribu7on) with k degrees of freedom. This is denoted by

⎟⎟⎠

⎞⎜⎜⎝

⎛=

kYXZ

X Y

X Y

Z

ktZ ~

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t distribu7on -‐ Uses

•  Very useful in – Hypothesis tes7ng – Construc7on of confidence intervals •  For the mean of a normal distribu7on based on a random sample.

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t distribu7on – Density Func7on

•  The density of t distribu7on with degrees of freedom is given by

21

2

1

2

21

)(

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠

⎞⎜⎝

⎛Γ

⎟⎠

⎞⎜⎝

⎛ +Γ

=

ν

νννπ

νttf

ν

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t distribu7on -‐ Density

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t distribu7on -‐ Shape

•  It is a bell-‐shaped distribu7on like normal distribu7on.

•  It is symmetric about 0. •  It has facer tails than normal distribu7on. •  t distribu7on comes closer and closer to normal distribu7on as the degrees of freedom get larger and larger.

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t distribu7on Mean and variance

•  Let have a t distribu7on with k degrees of freedom.

•  The mean of = 0 •  The variance of = when

Z

Z

2−kk 2>kZ

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F distribu7on •  Suppose and are independently distributed as chi-‐square

random variables with degrees of freedom m and n respec7vely.

•  Define •  The distribu7on of is called F distribu7on with degrees of

freedom m for the numerator and n for the denominator. This is denoted by

X Y

( )( )nYm

XZ =

Z

nmFZ ,~

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F distribu7on -‐ Uses

•  Very useful in: a)  Hypothesis tes7ng and Construc7on of confidence intervals for the

ra7o of variances of two normal distribu7ons based on independent random samples,

b) Hypothesis tes7ng of equality of means of several normal distribu7ons with the same variance.

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Density of F distribu7on,

•  Let

Where is a Beta func7on.

nmF ,

nmFX ,~

0,1

2,

2

1)(21

22

≥⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛

⎟⎠

⎞⎜⎝

⎛=

+−

−xx

nmx

nm

nmBxf

nmm

m

(.,.)B

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Density of F distribu7on, 21 ,ddF

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Proper7es of F distribu7on

•  Let •  Mean of =

•  Variance of =

• 

nmFU ,~

U 2,2

>−

nnn

U 4,)4()2()2(2

2

2

>−−

−+ nnnnnmn

mnFU ,~1

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Distribu7on Tables •  The probability tables for each of these distribu7ons have been calculated and tabulated for several values of Degrees of Freedom

•  We can find these tables online or in text books •  An easier way is to make use of sta7s7cal sohware that readily calculate these values for us

•  Using R: –  Chi-‐square: > pchisq(x,df) –  t-‐distribu7on: > pt(x,df) –  F-‐distribu7on: >pf(x,df1,df2)

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Thank You

(15) chi-square, student’s t and snedecor’s f distributions

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