1. (10 points) Let f(x) = S1 (+2) - 1, 2<1 1 - 4 sin(), >1 and g(x) = 5-24 (a) Sketch as much off and g as will fit on the graph below. Draw them in two different (dark) colors or one solid/one dashed, and clearly label the graph. -6 -4 -2 2 4 6 8 10X (b) Find the exact value of the limit below, and explain your reasoning in complete sentences. lim (g(x)) =
22 points possible 7/22 answered Question 14 < > ✓-4-2 +4 if << -5 Let f(2)= if x = -5 3x + 20 if 2> - 5 Calculate the following limits. Enter "DNE" if the limit does not exist. lim f(a) I-5 lim 2-5 lim f(x) = > Next Question 21 MacBook Air
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Consider the following. 1, -LSX<0. 10. OSX<L; f(x + 2) = f(x) (a) Sketch the graph of the given function for three periods. (In these graphs, L = 1.) f(x) — — - - - 1 -3 -2 -1 1 2 -3 3 3 -2 -1 . 2 1 (b) Find the Fourier series for the given function. R0 - 4 - ŠOx)
2. Let X and Y be continuous random variables having the joint pdf f(x,y) = 8xy, 0 <y<x<1. (a) Sketch the graph of the support of X and Y. (b) Find fi(2), the marginal pdf of X. (c) Find f(y), the marginal pdf of Y. () Compute jx, Hy, 0, 0, Cov(X,Y), and p.
Question 7 (5 points) Let f(x) = 24 and -2x, x < 5 9 3 22, x > 5 Evaluate(gof)(7) A/
Given, f(x) = f(x)= 4,0<x< 2 lx + 1, 2 < x < 4 (a) Sketch the graph of f(x) and its even half-range expansion. Then sketch THREE (3) full periods of the periodic function in the interval -12 < x < 12.
7. Let X be a random variable with density f(x) = 2/32 for 1<x<2, f(x) = 0 otherwise. Find the density of x2
x +1 if-2<x<3 |-x if x 23 8. Graph the function f(x)-
Given, f(x) = {x #1, 2 5x<4 4,0<x< 2 (a) Sketch the graph of f(x) and its even half-range expansion. Then sketch THREE (3) full periods of the periodic function in the interval -12 < x < 12. (6 marks) (b) Determine the Fourier cosine coefficients of Ql(a). (10 marks)