Unformatted text preview: f ( x ) and find the cdf F ( x ). (b) (10 points) If we use Unif(0 , 1) as the trial distribution for rejection sampling from f ( x ), what will be the maximal acceptance rate that we can achieve? (c) (10 points) Calculate the inverse function of F ( x ). Denote the inverse function by F-1 ( u ) for u ∈ [0 , 1]. (d) (10 points) Let X = F-1 ( U ), where U ∼ Unif(0 , 1). Find the expectation of X . 3 3. (30 points) Suppose that the joint density of two random variables X and Y is f ( x, y ) = 1 /π if x 2 + y 2 ≤ 1; otherwise. (a) (10 points) Let R and Θ be the polar coordinates of ( X, Y ). Find the joint density of ( R, Θ). (b) (10 points) What is the marginal distribution of R and what is the marginal distri-bution of Θ? Are Θ and R independent? Why or why not? (c) (10 points) Based on (a) and (b) design a method to draw uniform points within the unit circle. 4...
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- Spring '14