(15) Chi-square, Student’s t and Snedecor’s F distributions
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Transcript of (15) Chi-square, Student’s t and Snedecor’s F distributions
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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
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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
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Applied Sta+s+cs and Compu+ng Lab
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.
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Applied Sta+s+cs and Compu+ng Lab
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
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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
Density of Chi-‐square distribu7on with k degrees of freedom
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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
‘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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
t distribu7on -‐ Density
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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
Density of F distribu7on, 21 ,ddF
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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
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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Applied Sta+s+cs and Compu+ng Lab
Thank You