Question

4. Let X have the following PDF: sin(x) , 0 < x < π , otherwise Ix(x) = 0 Find the CDF of X Using the Probability Integral Transformation Theorem, describe the process of generating values from the density of X Using R, generate 1,000 values using your process in part b. Produce a histogram of these generated values, and overlay the density curve of X over top. (Hint: in R, the function acos calculates the inverse cosine function.) Using the transformation Y = cos(X), find the PDF of Y (Hint: for-1 < y < 1, you can use the identity sin(cos-1 y) = v 1-y2 ) Transform the generated values from part c using the transformation Y = cos(X). Produce a histogram of the transformed y values, and overlay the density curve of Y over top. a. b. c. d. e. If X ~ N(1,02), then the PDF of X is (x - H)2 for x E R.

Answer 1

" P(X lesthanorequalto x) = Integral_0^x 1/2 sin y dy F(x) = 1/2 [1 - cos x] (b) For generating random variable from density of X Take X = F^- 1 (y) where ~ Unit (0.1) y = 1/2 (1 - cos x) x = cos^- 1 (1 - 2y) - 1 lessthanorequalto 1 - 2y lessthanorequalto 1 Rightarrow 0 lessthanorequalto x lessthanorequalto pi (c) The code for generating values in R will be a = r unit(1000) b = a cos(1 - 2 times a) (b, freq = ""FALSE"") lines (b, sin (b)/2, t_y = ""p"") (d) f_ (y) = d/dy P(lesthanorequalto y) = d/dy p(cos X lessthanorequalto y) = d/dy p[x greaterthanorequalto cos^- 1 (y)] = d/dy Integral_cos^- 1 (y)^pi 1/2 sin x dx \r\n Thus PDF of y ~ (- 1, 1) (e) c = cos (b) list (c, freq = FALSE) lines (c, (1/2, 1000), t_y = ""p"")"