If W is a standard Brownian motion, find the distribution of W(t) + W (s).

Calculate the residual of P = (1.2, 2.4) with respect to the line 3x + 4y = 12.

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…

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Derive an asymptotic distribution for the MOM estimator ˜θ = 2X¯ of the form.(b) From this choose an approximate pivot and interval to get an interval that asymptotically has100(1 − α)% coverage.