 # mathematical statistics problems Anonymous

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MATH448.01 HW1 Spring 2018 Due: 02/09/2018 Textbook problems: p. 394-396, 8.6, 8.7, 8.8, 8.9, 8.13, 8.14, 8.16, 8.18, 8.19 Non-textbook problems: 1. Let X1 , X2 , ...., X2n−1 be iid exp (β) random variables. Note that there are an odd number of them. Let θ be the median of the exp (β) distribution. a. Calculate θ. b. Consider the following two estimators for θ. θb = X(n) and θe = X ln 2 Calculate the Mean Square Error (MSE) for both estimators. Which estimator has the larger MSE? # \$ 2. Let X1 , X2 , X3 , ..., Xn be iid random variables with E X14 < 1. Let E (X1 ) = µ and σ 2 = V ar (X1 ). Define S2 = n X # i=1 Xi − X \$2 n−1 a. Show that S is an unbiased estimator for σ 2 . b. Show that 2 # \$ E (X1 − µ)4 σ 4 (n − 3) V ar S 2 = − n n (n − 1) HINT: Note that 1 X (Xi − Xj ) S 2 = # n\$ 2 2 i Purchase answer to see full attachment

Ace_Tutor
School: Cornell University  you are welcome, and attached is my solution

Question 1:
(a) Suppose that X 1 , X 2 ,..., X 2 n 1 are iid exp    random variables and let  be the median of

exp    . Since N  2n 1is odd, the median is the n -th value. Since the number of X i   is
indeed a binomial random variable Y   with both parameters N and p  e  . Therefore,

  P  Median   
 P Y    n 
N
N i
N
    ei 1  e 
i n  i 
2 n 1 2n  1

 i
 2 n 1i
 
 e 1  e 
i 
i n 

(b) For the 1st estimator   X  n  , the Mean Square Error (MSE) is calculated by

MSE  E   

2


 Var     E    

 Var     2n  1 e
 Var    Bias 

2

2  2n  ...

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Tutor went the extra mile to help me with this essay. Citations were a bit shaky but I appreciated how well he handled APA styles and how ok he was to change them even though I didnt specify. Got a B+ which is believable and acceptable. Brown University

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