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