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please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details please answer non-text problems in details

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
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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

ei 1 e

i n i

2 n 1 2n 1

i

2 n 1i

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.

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