a4su112131 (1) - Y where P Y = θ = 1 Question 5(5 marks In...

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MATH 2131 3.00 MS2 Assignment 4 Total marks = 40 Question 1: Let r.v. X have pdf f ( x ) = ( 2 /x 3 , x > 1 , 0 , otherwise. (a) (3 marks). Compute the first two moments of X . (b) (1 mark). Except for the first moment, what can you say about all the other moments of X ? Question 2: Suppose that X N ( μ X , σ 2 X ) and Y N ( μ Y , σ 2 Y ) and that X and Y are independent. (a) (6 marks). Find the moment-generating function (mgf) of X . (b) (4 marks). Use mgf’s to find the distribution of aX + bY . Question 3: (3 marks). Let X 1 , X 2 , . . . be an infinite sequence of discrete r.v.’s such that p X n ( x ) = 1 / 2 n, x = ± n, 1 - 1 /n, x = 0 , 0 , otherwise. Let Y be a degenerate r.v. where P ( Y = 0) = 1. Show that the sequence X 1 , X 2 , . . . converges in probability to Y . Question 4: (5 marks). Let X 1 , X 2 , . . . be i.i.d. Uniform[0, θ ] r.v’s. Let Y n = max( X 1 , X 2 , . . . , X n ). Show that the infinite sequence of r.v.’s Y 1 , Y 2 , . . . converges in probability to the degenerate r.v.
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Unformatted text preview: Y where P ( Y = θ ) = 1. Question 5: (5 marks). In a grocery store checkout line, it has been found that the average time to process a customer is a r.v. X with a mean of 5 minutes and a standard deviation of 2 minutes. Use the CLT to approximate 1 the probability that 50 customers can go through the checkout line in less than 4 hours. Question 6: (8 marks). Let Y ∼ Uniform[0 , 1], and let X n = Y n . Show that { X n } converges with probability 1 to the degenerate r.v. X where P ( X = 0) = 1. Question 7: (5 marks). Let X 1 ,X 2 ,... be an infinite sequence of continu-ous r.v.’s such that f X n ( x ) = ( ( n + 1) x n , < x < 1 , , otherwise. Show that { X n } converges in distribution to the degenerate r.v. X where P ( X = 1) = 1. 2...
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