Probability theory 2008 Outline of lecture 5 The multivariate normal distribution  Characterizing properties of the univariate normal distribution  Different definitions of normal random vectors  Conditional distributions  Independence  Cochran’s theorem

Probability theory 2008 The univariate normal distribution - defining properties  A distribution is normal if and only if it has the probability density where   R and  > 0.  A distribution is normal if and only if the sample mean and the sample variance are independent for all n.

Probability theory 2008 The univariate normal distribution - defining properties  Suppose that X 1 and X 2 are independent of each other, and that the same is true for the pair where no coefficient vanishes. Then all four variables are normal. Special case: rotations other than multiples of 90 degrees x1x1 x2x2

Probability theory 2008 The univariate normal distribution - defining properties Let F be a class of distributions such that X  F  a + bX  F Can F be comprised of distributions other than the normal distributions? cf. Cauchy distributions

Probability theory 2008 The multivariate normal distribution - a first definition  A random vector is normal if and only if every linear combination of its components is normal Immediate consequences: Every component is normal The sum of all components is normal Every marginal distribution is normal Vectors in which the components are independent normal random variables are normal Linear transformations of normal random vectors give rise to new normal vectors

Probability theory 2008 Illustrations of independent and dependent normal distributions

Probability theory 2008 Illustrations of independent and dependent normal distributions

Probability theory 2008 Parameterization of the multivariate normal distribution  Is a multivariate normal distribution uniquely determined by the vector of expected values and the covariance matrix?  Is there a multivariate normal distribution for any covariance matrix?

Probability theory 2008 Fundamental results for covariance matrices Let  be a covariance matrix. Since  is symmetric there exists an orthogonal matrix C ( C’C = C C’ = I ) such that C’  C = D and  = CD C’ where D is a diagonal matrix. Since  is also nonnegative-definite, there exists a symmetric matrix B such that B B =  If X has independent components with variance 1, Y = BX has covariance matrix 

Probability theory 2008 The multivariate normal distribution - a second definition  A random vector is normal if and only if it has a characteristic function of the form where  is a nonnegative-definite, symmetric matrix and  is a vector of constants Proof of the equivalence of definition I and II: Let X  N( ,  ) according to definition I, and set Z = t’X. Then E(Z) = t’u and Var(Z) = t’  t, and  Z (1) gives the desired expression. Let X  N( ,  ) according to definition II. Then we can derive the characteristic function of any linear combination of its components and show that it is normally distributed.

Probability theory 2008 The multivariate normal distribution - a third definition  Let Y be normal with independent standard normal components and set Then provided that the determinant is non-zero.

Probability theory 2008 The multivariate normal distribution - a fourth definition  Let Y be normal with independent standard normal components and set Then X is said to be a normal random vector.

Probability theory 2008 The multivariate normal distribution - conditional distributions  All conditional distributions in a multivariate normal vector are normal  The conditional distribution of each component is equal to that of a linear combination of the other components plus a random error

Probability theory 2008 The multivariate normal distribution - conditional distributions and optimal predictors  For any random vector X it is known that E(X n | X 1, …, X n-1 ) is an optimal predictor of Xn based on X 1, …, X n-1 and that X n = E(X n | X 1, …, X n-1 ) +  where  is uncorrelated to the conditional expectation.  For normal random vectors X, the optimal predictor E(X n | X 1, …, X n-1 ) is a linear expression in X 1, …, X n-1

Probability theory 2008 The multivariate normal distribution - calculation of conditional distributions  Let X  N (0,  ) where Determine the conditional distribution of X 3 given X 1 and X 2  Set Z = a X 1 + bX 2 + c Minimize the variance of the prediction error Z - X 3

Probability theory 2008 The multivariate normal vector - uncorrelated and independent components The components of a normal random vector are independent if and only if they are uncorrelated

Probability theory 2008 The multivariate normal distribution - orthogonal transformations  Let X be a normal random vector with independent standard normal components, and let C be an orthogonal matrix.  Then Y = CX has independent, standard normal components

Probability theory 2008 Quadratic forms of the components of a multivariate normal distribution – one-way analysis of variance Let X ijij, i = 1, …, k, j = 1, …, n i, be k samples of observations. Then, the total variation in the X -values can be decomposed as follows:

Probability theory 2008

Decomposition theorem for nonnegative- definite quadratic forms Let where Then there exists an orthogonal matrix C such that with x = Cy (y = C’x)

Probability theory 2008 Decomposition theorem for nonnegative-definite quadratic forms (Cochran’s theorem) Let X 1, …, X n be independent and N(0;  2 ) and suppose that where Then there exists an orthogonal matrix C such that with X = CY (Y = C’X) Furthermore, Q 1, …, Q p are independent and  2  2 -distrubuted with r 1, …r p degrees of freedom

Probability theory 2008 Quadratic forms of the components of a multivariate normal distribution – one-way analysis of variance Let X ijij, i = 1, …, k, j = 1, …, n i, be independent and N( ,  2 ). Then, the total sum of squares can be decomposed into three quadratic forms which are independent and  2  2 -distrubuted with 1, k-1, and n-k degrees of freedom

Probability theory 2008 Exercises: Chapter V 5.1, 5.2, 5.6, 5.8, 5.14, 5.16, 5.17, 5.27