[試題] 105-2 鄭明燕 統計學導論 期中考
課程名稱︰統計學導論
課程性質：數學系選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/4/27
考試時限（分鐘）：110
試題 :
β
1. Let X be a random variable with cdf F(x) = 1 - exp(-αx ) for x ≧ 0
and F(x) = 0 for x < 0, where α > 0 and β > 0.
(a) (3 pts.) Show that F is a cdf.
(b) (3 pts.) Find the density of X.
(c) (4 pts.) Find the probabilities P(1 ≦ X < 2) and P(X = 3).
2. Let X and Y have the joint density
6 2
f(x,y) = ---(x+y) , 0 ≦ x ≦ 1, 0 ≦ y ≦ 1.
7
(a) (8 pts.) Find the two conditional densities.
(b) (12 pts.) Find E(Y|X = x) and Var(Y|X = x), 0 ≦ x ≦ 1, and vertify
that E[Var(Y|X)] < Var(Y).
3. A six-sided die is rolled 240 times.
(a) (6 pts.) Approximate the probability that the face showing a three
turns up between 30 and 60 times.
(b) (6 pts.) Approximate the probability that the sum of the face values
is greater than 400.
4. (8 pts.) If X and Y are independent exponential random variables with
λ = 1, find the distribution of X/Y.
5. (10 pts.) Let X , ..., X be a simple random sample without replacement
1 n
_
from a finite population {x , ..., x }. Is X an unbiased estimator of
1 n
2
μ , where μ is the population mean? If not, what is the bias?
6. Let X , ..., X be i.i.d. uniform on [0,θ].
1 n
(a) (6 pts.) Find the method of moments estimate of θ and its mean and
variance.
(b) (8 pts.) Find the MLE of θ and its mean and variance.
(c) (6 pts.) Compare the bias, variance and mean squared error of the MLE
to those of the method of moments estimate.
7. Let X , ..., X be i.i.d. random variables with the density function
1 n
θ
f(x|θ) = (θ+1)x , 0 ≦ x ≦ 1.
(a) (5 pts.) Find the MLE of θ.
(b) (5 pts.) Find the asymptotic variance of the MLE.
(c) (5 pts.) Find a sufficient statistic for θ.
(d) (5 pts.) Is the MLE a UMVUE? Why or why not?
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