課程名稱︰高等統計推論(上) 課程性質︰必修 課程教師︰江金倉 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰97/11/11 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Let (Ω,A*,P) denote the probability space and A_1,...,A_n belong to A.(註1) n n (1a) Show that P(∩A_i) = ΣP(A_i) - (n-1). i=1 i=1 n (1b) If P(A_i)=1 for all i, show that P(∩A_i) = 1. i=1 (1c) Let B belong to A* with P(B)>0. Show that P_B(A)=P(A|B) for all A belong to A* is a probability function. ∞∞ ∞ (1d) Show that P(∩∪A_n) = 0 if ΣP(A_n)<∞. k=1n=k n=1 2. Let X be a random variable with Var(X) = 0. Show that P(X = E[X]) = 1. 3. Let X ~ Gamma(n,1), W = (X-E[X])/√(Var(X)), and Φ(z) be the cumulative distribution function of a standard normal distribution. (3a) Show that lim P(W≦z) = P(Z≦z) for all z. n→∞ (3b) Show that((√n)/Gamma(n))*(z√n + n)^(n-1)*exp(-z√n - n) ≒ 1/ (√2π)*exp(-z^2/2) and z=0 gives Stirling's formula. 4. Let N_t denote the number of events occuring within the time period [0,t] and T be the time between two successive events. (4a) Write the nessssary conditions so that N_t follows a Poisson distribution with rate λ. (4b) Derive the probability density functions of N_t and T. 5. Let X ~ Poisson(θ),Y ~ Poisson(λ) are independent random variables, Z=X+Y. Find the distribution function of X condition on Z=z.(註2) 6. Let X_1,...,X_p be independent random variables and U_i = g_i(X_i) with g_i being a measurable function, i=1,...,p. Show that U_i's are mutually independent. 7. Let X_i ~ Gamma(α_i,β), i=1,...,m, be mutually independent. Derive the m distribution of W = ΣX_i. i=1 8. Assume that a family of p.d.f.s {f_T(t|θ):θ belong to Θ} has monotone likekihood ratio(non-decreasing case) in T. Show that P(T >t_0|θ_2) ≧ P(T >t_0|θ_1) for all θ_2 > θ_1. 註1: A* 是sigma algebra的符號,不過這裡好像打不出來. 註2: 這題是老師修改題目後寫在黑板上的，所以可能會有點出入. --

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