課程名稱︰機率導論
課程性質︰數學系必修
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︰2008年01月16日
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
Answer question in order. All random variables are given on probability
space(Ω,F,P).
1. Let (X,Y) be a random point that is uniformly distributed over the triangle
formed by the points (0,0),(2,0),and (0,3).
(a) Calculate E[X|Y], E[Y|X], E[X] and E[Y].
(b) Calculate var(X|Y), var(Y|X), var(X) and var(Y).
2. (a) Find the moment generating functions(transforms) of Bernouli and
Poisson random variables, respectively.
(b) Find the moment generating functions(transform) of the sum of a
Poisson-distributed number of independent, identically distributed
Bernouli random variables. What is the probability distribution of
the resulting sum?
3. Let U and V be independent standard normal random variables, and
X=3U+4V, Y=5U+aV, a belong to R. Let (σ_x)^2 and (σ_x)^2 be the variance
of X and Y, respectively, and ρ be the correlation cofficient of X and Y.
(a) Evaluated σ_x,σ_y, andρ.
(b) Find a so that X and Y are independent.
(c) Assume that a=12. Find the joint probabilty density function
f_X,Y of X and Y.
(d) Assume that a=12. Find the conditional density function
f_X|Y of X and Y.
4. Consider the following types of convergence for random variables X, X_1,
X_2,... When solving the questions, you may use general p>0 or any specific
p, for example, p=1 or p=2.
[A] The sequence X_n converges to X in L_p, p>0.
[B] The sequence X_n converges to X in probability.
[C] The sequence X_n converges to X with probability 1(almost surely).
[D] The sequence X_n converges to X in distribution(weakly).
(a) State the definition of each type of convergence listed above,
respectively.
(b) Discuss the relation between convergence in L_p ([A]) and convergence
in probability ([B]). That is, give a proof if (A)([B]) implies [B]
([A],respectively), or give a counterexample if it does not.
(c) Discuss the relation between convergence in probability ([B]) and
convergence with probabilty one([C]) as in (b).
5. Let Z_n, n=1,2,..., be a sequence of independent,idendically distributed
random variables. Write down the complete statments, including assumptions
and conclusions, of the following limit theorems of(Z_n). No proof is
needed.
(a) Weak law of large numbers;
(b) Strong law of large numbers;
(c) Central limit theorem.
6. Let W_1,W_2,... be independent, identically distributed random variables
with E[|W_1|]=μ<∞. Let Y_n=(W_n)/n^2, n=1,2,...
∞
(a) Show that Σ Y_n converges absolutely with probability 1. Find the
n=1
almost sure limit of the series.
(b) Based on the above observation, find the almost sure limit of Y_n,and
(Y_1+Y_2+....+Y_n)/n as n→∞, respectively.
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