PrcMidterm102C_19F.pdf - Stats 102C Practice Midterm Exam Name UID There are 3 questions for a total of 100 points Please include necessary intermediate

PrcMidterm102C_19F.pdf - Stats 102C Practice Midterm Exam...

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Stats 102C Practice Midterm Exam Name: UID: There are 3 questions for a total of 100 points. Please include necessary intermediate results. Use only the paper provided. You may write on the back if you need more space, but please indicate on the front. Question Points 1 2 3 Total 1

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1. (30 points) Let f ( x ) and g ( x ) be two normalized densities on R , and g ( x ) > 0 for all x . Suppose that we only know f ( x ) q ( x ) and want to use importance sampling to estimate the normalizing constant Z q = R q ( x ) dx with g ( x ) as the trial distribution. Define w ( x ) = q ( x ) g ( x ) as the importance weight. (a) (10 points) Show that Z q = E g [ w ( X )]. (b) (10 points)Suppose that Var g [ w ( X )] = σ 2 , where Var g denotes variance with respect to the distribution g ( x ). Let ˆ Z q = 1 n n i =1 w ( X ( i ) ), where X (1) , · · · , X ( n ) are i.i.d. samples from g ( x ). Find the expectation and the variance of ˆ Z q . (c) (10 points) Which of the following will be an accurate estimate of E f ( X ) = R x f ( x ) dx as the sample size n → ∞ ? Why? (A) 1 n n i =1 X ( i ) ; (B) 1 n n i =1 X ( i ) w ( X ( i ) ); (C) n i =1 X ( i ) w ( X ( i ) ) n i =1 w ( X ( i ) ) . 2
2. (40 points) Let f ( x ) (1 - x ) be a probability density on the interval [0 , 1]. (a) (10 points) Normalize density

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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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