16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde V-M yr y Y=VMT Y=VMV! 1…Vn 1→ V…. y is the covariance matrix for the random variable y= vy so the a are the variance of the Yi. n n These are the principal coordinate with intercepts at y, =+ca with athe standard deviation of yi lote that two random variables, each having a normal distribution singly, do not necessarily have a binormal ioint distribution However, if the random variables are independent and normally distributed, heir joint distribution is clearly a multidimensional normal distribution 9/30/2004955AM Page 9 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 9 of 10 1 11 1 1 11 1 1 1 1 1 1 ... ... ... T TT T T T n T n n T n xM x yV M V y yY y Y V MV Y VMV VM v v v v v v λ λ λ λ − − −− − − − −− = = = = ⎡ ⎤ ↑ ↑ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ↓ ↓ ⎣ ⎦ ⎡ ⎤ ←→↑ ↑ ⎡ ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ←→↓ ↓ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ O Y is the covariance matrix for the random variable Y VX = , so the λi are the variance of the Yi. 1 1 1 1 n Y λ λ − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ O 1 2 22 2 1 2 1 2 1 2 1 1 ... T n T n n y yc yy y yY y c λ λ λλ λ − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = + ++ = O These are the principal coordinate with intercepts at i i y c = ± λ with λi the standard deviation of yi. Note that two random variables, each having a normal distribution singly, do not necessarily have a binormal joint distribution. However, if the random variables are independent and normally distributed, their joint distribution is clearly a multidimensional normal distribution