[試題] 107-2 張志中 機率導論 期中考
課程名稱︰機率導論
課程性質︰必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰108/5/2
考試時限（分鐘）：110分鐘
試題 :
1.(5+10+5=20 points) Let S and T be independent and absolutely continuous random
variables with density functions f and f , respectively. Let Z = S+T.
S T
(a) Show that f (z|s)=f (z-s).
Z|S T
(b) Assume that S and T are exponentially distributed with parameter λ>0.
Find/Recognize the conditional density f (s|z) of S, given Z=z.
S|Z
(c) Evaluate E[S|Z].
2.(8+6+6=20 points) Let X, Y be independent random variables having exponential
distribution with common parameter λ>0. Put U=X+y, V=X/(X+Y).
(a) Find the joint density f of(U, V).
U,V
(b) Find/Recognize the individual probability laws of U and V, respectively.
Use the Table in Question 5 if necessary. Determine if U and V independent
or not.
(c)Evaluate E[X|V], and then E[Y|V].
X
Hint: First consider E[X|V=v]=v E[—|V=v]. Use the Table in Question 5 if
v
necessary.
3. (10+8+12=30 points) Let R , ..., R be independent absolutely continuous
1 n
random variables with common distribution function F and density f. Let T , ...,
1
T be the order statistics of R , ..., R such that T <T <...<T a.s.
1 n 1 2 n
(a) Find the joint distribution and density functions of (T , T -T ).
n-1 n n-1
(b)Find the (individual) density functions of T and T -T , respectively.
n-1 n n-1
*註1
(c)Now suppose that R , R , ..., R are independent exponentially distributed
1 2 n
random variables with parameter λ>0.
(i) Write the answers of (a) and (b) explicitly, and determine if T and
n-1
T -T are independent or not.
n n-1
(ii)Evaluate E[T |T ].
n n-1
*註2
4.(10*3=30 points) Fix a random variable R defined on a probability space (Ω,
F, P) and a Borel set A∈B(E) which has positive Lebesgue measure. Let 1 (R)
A
be the random function that 1 (R)(ω)∈A, and 0 otherwise. Here E is the set
A
of real numbers. Assume that P{ω:R(ω)∈A}=α > 0.
(a) Explain why 1 (R) is a random variable, and describe the σ-field σ(1 (R))
A A
generated by 1 (R).
A
(b) Let R , R , ... be a sequence of independent and identically distributed
1 2
random variables such that each R has the same distribution as R.
j
Let Y =exp(1 (R )), j=1, 2, .... It is clear that Y , Y , ... are random
j A j 1 2
variables, and are independent and identically distributed. Prove, by
1
direct computation, that —(Y +...+Y ) converges to some limit m∈E in L as
n 2
n→∞. Determine the limit m.
(c) Apply appropriate inequalities to conclude from (b) that the convergence
also holds in L and in probability as n→∞.
1
5.(2*10=20 points)填表題，要填常見的distribution的density function、mean和
variance。印象中有exponential、Normal、Binomial、Poisson，然後Gamma直接給了。
*註1：解答裡沒有把積分展開
*註2：(c)的(ii)，原題為「Evaluate E[T |T ]」，所以送了6分
n-1 n
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