STATISTICS 4M03/6M03: Test #1
30 October 2003
Students may bring and use any calculators, tables, books or notes.
[10] 1. Let y1, y2, …, yn be iid N(0, 1). Define zi = yi+1 – yi, i = 1, …, n-1, and z = (z1, …, zn-1)'. Find the distribution of z.
2. Consider a bivariate random variable (X, Y) with a distribution that is uniform over a triangle defined by the vertices (0, Ö3), (-1, 0), (1, 0).
[15] (a) Find and graph the marginal and conditional distributions. Find the mean vector and the covariance matrix. Are X and Y statistically independent? Find the regression lines. Draw a graph to show the triangle and the regression lines. Draw a 95% triangle of concentration.
[10] (b) Consider a bivariate normal distribution with the same mean vector and the same covariance matrix as the distribution in (a). Add its 95% ellipse of concentration to the graph you drew in (a).