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)


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

X1,X2,...,Xn ~ N(μ,σ^2) =>

Xbar ~ N(μ,σ^2/n) =>


Z = 

   Xbar-μ

   ----------

   σ/root(n)


~ N(0,1)



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

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)



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

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

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

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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