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
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