Homework 11/1 October 27 2011 1. Let Xi; i = 1; :::; n be i.i.d. N(; 2) and Sn =

Homework 11/1 October 27 2011 1. Let Xi; i = 1; :::; n be i.i.d. N(; 2) and Sn = X1 + ::: + Xn. Find the conditional distibution of X1; :::;Xn given Sn. 2. a. Suppose Xi are i.i.d. N(01) and O is an orthonormal matrix that is O0O = I where I is the identity matrix. Show that the components of Y = OX are also i.i.d. N(01). b. In particular in two dimensions show that the rotation matricescos() sin() ??sin() cos()are orthogonal. Conclude that the angle formed by the origin and (X1;X2) is uniformly distributed on an appropriate interval maybe (0; 2). i.e.
Homework 11/1 October 27 2011 1. Let Xi; i = 1; :::; n be i.i.d. N(; 2) and Sn = X1 + ::: + Xn. Find the conditional distibution of X1; :::;Xn given Sn. 2. a. Suppose Xi are i.i.d. N(01) and O is an orthonormal matrix that is O0O = I where I is the identity matrix. Show that the components of Y = OX are also i.i.d. N(01). b. In particular in two dimensions show that the rotation matricescos() sin() ??sin() cos()are orthogonal. Conclude that the angle formed by the origin and (X1;X2) is uniformly distributed on an appropriate interval maybe (0; 2). i.e. tan??1(X2 X1 ) has a uniform distribution. Conclude that if X1;X2 are i.i.d normals with mean 0 then X2 X1 has a Cauchy distribution. Warning:draw the conclusion from the facts on the page do not rederive the distribution of X1 X2 . 3. 5.10.7 Suppose X1;X2 have a bivariate normal distribution for which E(X1jX2) = 3:7 ?? :15X2E(X2jX1) = :4 ?? :6X1 and 2(X2jX1) = 3:64. Find the mean and covariance matrix for (X1;X2). 4. 8.3.1 Assuming that X1; :::; :Xn form a random sample from a N(; 2) distribution show that the sample variance has a gamma distribution and nd its parameters. 5. How many terms of the form X2i X2j ; i 6= j occur when (X1 +:::+Xn)4 is expanded ? 1
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