Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.

Bighorn sheep are beautiful wild animals found throughout the western United States. Let x be the age of a bighorn sheep (in years), and let y be the mortality rate (percent that die) for this age group. For example, x = 1, y = 14 means that 14% of the bighorn sheep between 1 and 2 years old died. A random sample of Arizona bighorn sheep gave the following information: y USE SALT 2= Σχ = 15; Σy = 88.3; Σx2 = 55; Σy2 = 1,594.89; Σxy = 275.8 (a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to three decimal places.) X= y = b= ŷ= (b) Draw a scatter diagram for the data. Plot the least-squares line on your scatter diagram. y y 22 20 18 16 14 12 10 0 y 22 20 14.4 18 y 22 VX 20 18 16 14 12 10 5 1 (c) Find the sample correlation coefficient r and the coefficient of determination 2. (Round your answers to three decimal places.) 16 14 12 5 2 3 4 19.9 14.4 19.6 20.0 1 1 2 2 3 3 4 4 5 22 6 20 18…

N(u, E) be a Gaussian random variable with mean u E Rd and covariance matrix E e Rdxd. Assuming that E is invertible. Let a e Rd be a fixed vector, and Y = a'X € R be a random variable. Calculate E[Y], Var(Y), E[X|Y = y], and Cov(X|Y = y) = E[Xx'|Y = y] – E[X|Y = y]E[X"|Y = y]. Let X

Say what will be the expected value of the momentum (px) of a particle described by the wave function

Suppose that, as in Exercises 5.11 and 5.79, Y1 and Y2 are uniformly distributed over the triangle shaded in the accompanying diagram. 5.95 (0, 1) (-1, 0) (1, 0) a Find Cov(Y1, Y2). b Are Y1 and Y, independent? (See Exercise 5.55.) Find the coefficient of correlation for Y1 and Y2. d Does your answer to part (b) lead you to doubt your answer to part (a)? Why or why not?