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Material Type: Exam; Class: Cryptgrphy/Data Securty; Subject: COMP Computer Science; University: University of Memphis; Term: Fall 2001;
Typology: Exams
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(a) P (1/ 2 ≤ X). (b) P (X ≤ Y ). (c) Marginal p.d.f. of X and Y.
∑n i=1 Xi^ and^ S^2 =^
∑n i=1 X i^2.
(a) Define the central limit theorem for Y. (b) If f (x; μ, σ^2 ) is normal with μ = 0, show that the variable Z = {S^2 −nσ^2 }/{σ^2 (2n)−^1 /^2 } converges in distribution to N (0, 1) as n → ∞, where N (0, 1) denotes the normal distribution with mean 0 and variance 1.
(a) If X 1 ∼ P oisson(λ), X 2 ∼ P oisson(μ), and X 1 , X 2 are independent find P (X 1 = k|X 1 + X 2 = 2k) (b) If (X 1 , X 2 , X 3 ) ∼ Trinomial(n, p 1 , p 2 , p 3 ), that is,
P (X 1 = x 1 , X 2 = x 2 , X 3 = x 3 ) =
n! x 1 !x 2 !x 3!
px 11 px 2 2 px 3 3 ,
where p 3 = 1 − p 1 − p 2 , X 3 = n − X 1 − X 2 , find P (X 1 = k|X 1 + X 2 = m).
(a) 1, 2 (b) 2, 4 (c) 1, 2 , 2 , 4.
(a) Find the maximum likelihood estimate of λ. (b) Test H 0 : λ = 0.2 versus H 1 : λ > 0 .2.
f (x; θ) = 1/θ x^1 /θ−^1 , 0 < x < 1 , 0 < θ < ∞.
(a) Find the moment estimator of θ. (b) Find the maximum likelihood estimator of θ. (c) Find Rao-Cram´er lower bound of any unbiased estimator θˆ for θ.
i 1 2 3 4 5 6 7 xi 0 10 49 52 59 64 64 yi 12 20 60 43 37 58 63
In fact,
∑ 7 i=1 xi^ = 298,^
∑ 7 i=1 yi^ = 293,^
∑ 7 i=1 x^2 i = 16878, and^
∑ 7 i=1 xiyi^ = 15303. (a) Find the estimated regression line of Y on X: y = ˆα + βxˆ. (b) Plot the seven data points and the estimated regression line on the same graph. (c) If a similar student in a similar situation in the future obtained a homework score of 30, what would be the best linear estimate of his subsequent examination score?
(a) How large a sample size is needed to obtain an α-level most powerful test with power 1 − β? (Assume the sample size is large enough to apply the Central Limit Theorem.) (b) Compute the sample size in part (a) if λ 0 = 2, λ 1 = 1, α = 0.05 and 1 − β = 0.90.
