课程名称︰机率导论 课程性质︰必修 课程教师︰张志中 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰108/5/2 考试时限（分钟）：110分钟 试题 : 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 *注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 *注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)填表题，要填常见的distribution的density function、mean和 variance。印象中有exponential、Normal、Binomial、Poisson，然後Gamma直接给了。 *注1：解答里没有把积分展开 *注2：(c)的(ii)，原题为「Evaluate E[T |T ]」，所以送了6分 n-1 n --

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