# useful distributions

X1,X2,...,Xn ～ N(μ,σ^2) =>

Xbar = 1/nΣXi

χ^2

=

Σ(Xi-Xbar)^2

-------------------

σ^2

=

nS^2

----

σ^2

～ χ^2(n-1) (chi-squared distribution with n-1 degrees of freedom)

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X1,X2,...,Xn ～ N(μ,σ^2) =>

Xbar ～ N(μ,σ^2/n) =>

Z =

Xbar-μ

----------

σ/root(n)

～ N(0,1)

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X1,X2,...,Xm ～ N(μx,σx^2)

Y1,Y2,...,Yn ～ N(μy,σy^2) =>

Xbar ～ N(μx,σx^2/m)

Ybar ～ N(μy,σy^2/n) =>

Xbar-Ybar ～ N(μx-μy,σx^2/m+σy^2/n) =>

Z =

(Xbar-Ybar)-(μx-μy)

---------------------

root(σx^2/m+σy^2/n)

～ N(0,1)

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X1,X2,...,Xn ～ N(μ,σ^2) =>

T =

root(n-1)(Xbar-μ)

-----------------

S

～ t(n-1) (Student's t distribution with n-1 degrees of freedom)

used in test of μ

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X1,X2,...,Xm ～ N(μx,σx^2)

Y1,Y2,...,Yn ～ N(μy,σy^2)

σx^2 = σy^2

=>

T =

root(m+n-2){(Xbar-Ybar)-(μx-μy)}

--------------------------------------

root((1/m+1/n)(mSx^2+nSy^2))

～ t(m+n-2)

used in test of μx-μy

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