16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Two dimensional case- continued. my=Xx f(x1,x2) m12x1 11m2-ml2 2(m1m2-m2) m1=6 my pna In terms of these symbols PI f(x12x2)= 八02)(2 2(1-n2) ote that if a set of random variables having the multidimensional normal distribution is uncorrelated they are independent. This is not true in general 9/30/20049:55AM Page 10 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 10 of 10 Two dimensional case- continued: m XX ij i j = 2 2 22 1 12 1 2 11 2 212 2 2 11 22 12 11 22 12 2 11 1 2 22 2 12 12 12 12 2 1 12 2 12 1 12 2 212 2 1 2 12 1 2 ( , ) exp 2 2( ) In terms of these symbols: 2 1 ( , ) exp 2 1 x y m x m xx m x f xx mm m mm m m m m x xx x f xx π σ σ µ µ ρ σ σ ρ σ σσ σ πσ σ ρ ⎡ ⎤ − + = −⎢ ⎥ − ⎣ ⎦ − = = = = ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ − + ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ = − − 2 2 12 2(1 ) ρ ⎡ ⎤ ⎢ ⎥ ⎟ ⎢ ⎠ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Note that if a set of random variables having the multidimensional normal distribution is uncorrelated, they are independent. This is not true in general