課程名稱︰統計學 課程性質︰數學系選修、系統生資學程選修、神經認知學程必修 課程教師︰鄭明燕 開課學院：理學院 開課系所︰數學系 考試日期︰2012年04月16日 考試時限：15:30 - 18:20 是否需發放獎勵金：是 試題 : Statistics　　　　　　　　　　Midterm Examination　　　　　　　　 April 16,2012 1. (a) (6 pts.) Consider a sample space Ω with event space F. State the three 　　　 axioms of probability for a probability function P(.) defined on F. (b) (6 pts.) Prove thtat, for arbitrary events A and B in F, 　　　 P(A∩B)≧P(A)+P(B)-1. 　 (c) (8 pts.) Let A, B and C be mutally independent events. Explain what is 　　　 meant by this statement. Deduce from this assumption that A∩B and C are 　　　 independent events. 2. (10 pts.) Show that if the conditional probabilities exist, then 　　　　　　P(A1 ∩ A2 ∩ ... ∩An) 　　　　＝　P(A1)P(A2 | A1)P(A3 | A1 ∩ A2)...P(An | A1∩A2∩...∩A(n-1)). 3. (10 pts.) Suppose X~N(μ,σ^2) and Y=aX+b. Show that Y~N(aμ+b,(aσ)^2). 4. (10 pts.) Find the probability density function of Y=exp(Z), 　 where Z~N(μ,σ^2). 5. (10 pts.) If X and Y are independent random variables with finite variances, 　 find Var(XY) in terms of the means and variances of X and Y. 6. If U1,...,Un are independent Uniform(0,1) random variables. 　 (a) (12 pts.) Find the density functions of U(n) and U(1). 　 (b) (8 pts.) Find E( U(n) - U(1) ). 7. A six-sided die is rolled 200 times. 　 (a) (10 pts.) Approximate the probability that the face showing a two turns 　　　 up between 40 and 60 times. 　 (b) (10 pts.) Approximate the probability that the sum of the face values is 　　　 greater than 500. --

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