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2. Applying rejection sampling. Consider the target pdf: 5'0 1 . : : : . : . : - ' 06.00.1 0.20.3 0.4 0.5 0.6 0.7 0.8 0.9' 1.0 The following samples are generated iid from an uniform distribution: 0.8235, 0.4387, 0.6948, 0.3816, 0.3171, 0.7655, 0.9502, 0.7952, 0.0344, 0.1869. a. Use the samples to sample from the target distribution. In order to do this, separate the samples int( DW"'ad 0.8235, 0.4387), (0.6948, 0.3816), (0.3171, 0.7655), (0.9502, 0.7952), (0.0344, 0.1869) and then apply rejection sampling. b. Assume that the target pdf is only nonzero between 0.5 and 0.1. Find a way of generating more samples from the available uniform samples. (Hint: Transform some of the samples so that their pdf is restricted to [0.5, 1].) 0. Another possible way of increasing the number of accepted samples is reordering them to yield different pairs so that there are less rejections. What is the problem with this? 3. Presents. For Christmas a teacher of a class of 77. children asks their parents to leave a present under the Christmas tree Dowload'assroom. The day after each child picks a present at random. We are interested in computing the expected number of children that end up getting the present bought by their own parents. a. What is the pmf for the indicator random variable L; corresponding to the event kid 1' gets the present bought by their own parents? b. Are L; and Ij independent if 1' 7E j? Justify your answer. c. What is the expected number of children that end up getting the present bought by their own parents? 4. Motions. a. A xed amount a is placed in one envelope and an amount 5a is placed in the other. One of the envelopes is opened (each envelope is equally probable), and the amount X is observed to be in it. Let Y be the (unobserved) amount in the other envelope. Compute E () and E(Y)/E(X)- b. Let Xi, 1 S 73 S 3, be identically distributed random variables with E(X1) = 1. Find X1 E (X1+X2+X3 ) '