STAT2601

• Criterion 1

  • fx,y(x, y) ≥0
  • fy (y)>0 for yЄ Y()
  • fxy (aly) ≥0
• Jacobian matrix represents the new parameters with old ones
• Binomial distribution is a sum of Bernoulli distributions.
• Gamma distribution is a sum of exponential distributions.
• Independence

  1. can be factorised
  2. supports of X and Y do not rely on each other ?(rectangular)
• universality of the uniform, i.e., F(X) ~ U(0, 1), only applies to continuous distribution
• Correlation means linear association; if X,Y are independent then they must be uncorrelated, but not vice versa
• Corr(X,Y) = $\sigma_{X,Y} / \sigma_X \sigma_Y$
• Exp (1/2)=𝜒^2(2)
• X~𝛤(a,b); Y=cX

  ==> Y~𝛤(a,b/c)
• 𝜒^2(r) = 𝛤(r/2, 1/2)
• 𝜒^2(a )+𝜒^2(b )=𝜒^2(a+b )
• X, Y~N; W=aX+bY 

  ==> W~N(aux+buy, a^2σx^2+b^2σy^2)
• Beta (1,1)~U(0,1)