[試題] 103下 陳宏 機率導論 第三次小考

[xavier13540 (柊 四千)]

課程名稱︰機率導論
課程性質︰數學系大二必修
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/06/09
考試時限（分鐘）：60

試題 :

                                                                            2
1. (25 points) Let X be a normal random variable with mean 0 and variance σ .


   Let S be a random variable which is independent of X and such that P(S = 1) =


   P(S = -1) = 1/2.


   (a) (15 points) Show that SX is a normal random variable and determine its


       mean and variance. (You can use moment generating function to show it.)


   (b) (10 points) Show that X and SX are uncorrelated, but X and SX are not


       independent.


2. (25 points) Let N, X , X  be three independent random variables. Let Y =
                       1   2
    N
   Σ X  where N is a random variable taking values 1 and 2 with probability 1/2
   i=1 i
                                            2
   and 1/2 while E(X ) = μ and Var(X ) = σ . Determine E(Y) and Var(Y).
                    i                i

3. (25 points) A basketball team scores baskets according to a Poisson process


   with rate λ = 2 baskets per minute.


   (a) (10 points) What is the expected amount of time until the team scores its


       first basket?


   (b) (5 points) Given that at the five minute mark of the game the team has


       scored exactly one basket, what is the probability that the team scored


       the basket in the first minute?


   (c) (10 points) What is the probability that the team scores exactly three


       baskets in the first five minute of the game? (Computation of the


       numerical value is not necessary.)


4. (25 points) Show that a Poisson process is continuous in probability by


   arguing


                   P(N    - N  > η) = P(N    - N  ≧ 1) < ε
                      t+h    t            t+h    t

   Hint: Associate the distribution of N    - N  and the distribution of N .
                                        t+h    t                          h

5. (25 points) Distribute n balls independently at random into n boxes. Let N
                                                                             n

   be the number of empty boxes. Denote N  by the sum of I , ..., I , where I  =
                                         n                1        n         i

   1 if the i-th box is empty and 0, otherwise.


   (a) (4 points) Write N  in terms of I , ..., I .
                         n              1        n

   (b) (6 points) Derive E(I ) and Var(I ).
                            1           1

   (c) (10 points) Derive E(I I ) and Cov(I , I ).
                             1 2           1   2
                                               -1
   (d) (5 points) Show that N /n converges to e   in probability as n goes to
                             n

       infinity.


/*

1. 我修改了第4題的題敘，讓題意比較清楚。

2. 5(d)原本沒有"in probability"，這是考試中老師加上去的。

*/

--
                 2                         2    1                  ψxavier13540
給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system
╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則
╰x(0) = x°                     t
ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三
次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。

--
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※ 編輯: xavier13540 (140.112.249.76), 06/24/2015 11:27:27
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