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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