STA3702 Assignment 2 2024 (Unique Nr. 199414) - DUE 7 June 2024 ;100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on ...(.+.2.5.4.7.7.9.5.4.0.1.3.2)...........
Question 1 [44]
The number of cars, X , passing through a certain intersection every week ...
Question 1 [44]
The number of cars, X , passing through a certain intersection every week has Poisson dis-
tribution with mean λ. Let X 1, X2, ..., Xn be independent variables representing the number of
cars passing through the intersection on n randomly chosen weeks of the year. Furthermore,
let
1 X 1X
n n
2 n−1 2
S2 = Xi − X S⋆2 = S and X = Xi
n − 1 i=1 n n
i=1
be three competing estimators of λ.
E [X i− λ] 4 λ 2(n − 3)
Given: V (S ) =
2 − .
n n(n − 1)
(a) Determine: (i) E X ; and (ii) prove or disprove that E (S 2) = λ. (10)
Xn 2
Xn
Hint: Xi − X = X i2 − nX 2.
i=1 i=1
(b) Prove or disprove that S⋆2 is a method of moments estimator of λ.
Xn 2
Xn
Hint: X i − X = X i2 − nX 2. (6)
i=1 i=1
(c) Determine the bias of S⋆2
in estimating λ. (4)
(d) Determine the variances of: (i) X ; and (ii) S⋆2 . (8)
(e) Determine the mean square errors of: (i) X ; (ii) S2; and (iii) S⋆2 . (4)
(f) Which of X , S2 and S⋆2 is the least accurate estimator of λ. Justify your answer. (3)
(g) Prove or disprove that X , S2 and S⋆2 are consistent estimators of λ. (9)
Question 2 [42]
The lifetime ( X in years) of an electronic component manufactured by certain company has
a distribution with probability density function:
1 (x − µ)
θ exp − θ
if θ > 0 and x > µ ,
f (x|θ) =
0 otherwise;
where θ is unknown and µ is known. The company has hired you to estimateθ. Suppose that
X 1, X2, ..., Xn are the lifetimes of n randomly chosen electronic components manufactured by
the company.
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller LIBRARYpro. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R48,65. You're not tied to anything after your purchase.
