課程名稱︰統計學
課程性質︰數學系選修
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/4/27
考試時限（分鐘）：15:30~17:30
試題 :
Part 1: 200 points
1. (50 points) New laser altimeters can measure elevation to within a few
inches, without bias, and with no trend or pattern to the measurements. As
part of an experiment, 10 readings were made on the elevation of a mountain
peak. These averaged out to 73.631 inches, with a standard deviation of 10
inches. The following fact holds
_ 10 _ 10
P(X －1.96 ———<μ< X ＋1.96 ———) = 0.95.
10 √10 10 √10
_
(a) (25 points) In this problem, 73.631 is denoted by x . Does the
10
following statement holds?
_ 10 _ 10
P(x －1.96 ———<μ< x ＋1.96 ———) = 0.95.
10 √10 10 √10
_
Give reasons to support your answer. Think carefully on what is x in
10
the above statement. Here μ is a number that we don't know.
(b) (25 points) Explain the meaning of 0.95 in terms of frequency point of
view as follows:
Frequentist probability or frequentism is a standard interpretation
of probability; it defines an event's probability as the limit of
its relative frequency in a large number of trials.
2. (40 points) The university administration assures a mathematician that he
has only one chance in 10000 of being trapped in a much-maligned elevator in
the mathematical building. If he goes to work 5 days a week, 52 weeks a
year, for 10 years and always rides the elevator up to his office when he
first arrives, what is the chance that he will never be trapped? That he
will be trapped once? Twice? Assume that the outcomes on all the days are
mutually independent.(a debious assumption in practice)
3. (50 points) The density of a uniform random variable is
f(u) = 1, 0 < u < 1, f(u) = 0 otherwise.
Suppose we have seven independent uniforms U , U , ..., U and we order them
1 2 7
in terms of its magnitude and write them as
U ≦U ≦U ≦U ≦U ≦U ≦U .
(1) (2) (3) (4) (5) (6) (7)
(a) (30 points) Find the probability of P(x≦U ≦x+h, y≦U < y+k,
(2) (4)
1－l≦U ) in terms of x, y, h, k and l. Denote it by g(x,y,h,k,l).
(7)
Here, x+h < y < y+k < 1－l. 註："1－l" 前者是數字1，後者是英文字母l
(b) (20 points) Find the limit of g(x,y,h,l)/(hk) as both h and k go to
zero.
4. (60 points) Let X, Y, and Z be uncorrelated random variables with variance
2 2 2
σ ,σ ,σ , respectively. Find Cov(U,V) and ρ , where U=2X+3Y, V=-4X-Z.
X Y Z UV
Part 2: 180 points
5. (60 points) Consider the following model
2
T ~ Exponential(λ), Z ~ N(0,σ ),
Y|Z ~ Uniform(0,|Z|), X = TZ + Y,
and T is independent of (Z,Y).
(a) (30 points) Describe E(Y|Z) and find E(Z‧1({Z≧0})).
(b) (30 points) Derive E(X) and Var(X).
6. (60 points) The algorithm for generating a standard normal random variable Z
is
Step 1. Generate Y with an exponential distribution at rate 1; that is,
generate U and set Y = －ln(U).
Step 2. Generate U.
2
If U≦exp(-(Y-1) /2), set |Z|=Y; otherwise go back to Step 1.
Step 3. Generate U. Set Z = |Z| if U≦0.5, set Z = -|Z| if U > 0.5.
Provide an argument on showing that the above algorithm indeed leads to a
standard normal random variable Z.
7. (60 points) Consider a sequence of random variables with μ = E(X ) and
i i
n
σ = E(X -μ )(X -μ ). Write Y = Σ (X -μ )/n.
ij i i j j n i=1 i i
(a) (20 points) Show that E(Y ) = 0.
n
(b) (20 points) Find Var(Y ) in terms of σ .
n ij
|i-j|
(c) (20 points) When |σ | < Mr with 0 ≦ r < 1, use Chebyschev
ij
inequality to show that Y converges to 0 in probability.
n