課程名稱︰機率
課程性質︰必帶
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資訊系
考試日期（年月日）︰2012/06/18
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
說明：總共12題，每題10分，可按任何順序答題。歡迎使用事先準備的A4大抄。
每題難度不同，請審慎判斷恰當的答題順序。
第一題 The joint density function of random variable X and Y is
f(x,y)={ y^(-1) * e^[-(y + x/y )] , if x>0 and y>0
0 , otherwise.
Prove E[X]=E[Y].
第二題 延續上一題，請計算Cov(X,Y)。
第三題 證明Cov(X,E[Y|X])=Cov(X,Y).
第四題 請證明兩個彼此獨立的常態隨機變數X與Y相加所得的隨機變數X+Y仍然擁有
常態分佈。
第五題 Let X be a Poisson random variable with parameter λ. Prove (e.q.,
using Chernoff bound) that for any nonnegative integer i with i < λ,
we have P(X≦i) ≦ [e^(-λ) * (eλ)^i] / i^i
第六題 Let X1, X2, ... be a sequence of independent and identically
distributed random variables, each having expectation 0 and variance 1. For
any positive number ε, use Chebyshev's inequality to prove
lim P( |(X1+X2+...+Xn) / n| ≧ ε) = 0
n->∞
第七題 Consider the Markov chain X of gambler's ruin with four states
{0,1,2,3}, where 0 and 3 are the absorbing states. Standing on a
non-absorbing state i, the gambler has probability 0.75 to go to state
i+1 and probability 0.25 to go to state i-1 at the next time index.
Let X(0)=1. Please compute the expected number of time indices, including
time 0, in which X stays in state 1 or 2.
第八題 老鼠在九個節點的「田」字型的迷宮裡跑來跑去，每單位時間，從目前的位置
，根據接下來所描述的機率跑到相鄰的點上去：有左有右時左右移動的機率相同，有上
有下時上下移動的機率相同，有左右有上下時左右移動的總機率是上下移動之總機率
的兩倍。請證明這個跑來跑去的過程是一個time-reversible Markov chain。也請計算
長時間下來，老鼠在這九個節點的機率分別為何？
第九題 Let i and j be two distinct states of X such that Pn[i,j] * Pm[j,i]>0
holds for some positive integers m and n. Prove that if i is recurrent, then
j is also recurrent.
第十題 Let X ba an irreducible finite Markov chain with n states. For each
i = 1, 2, ... , n, let ri be the long-run proportion of state i. Prove that
the row vector r = (r1, r2, ..., rn) satisfies r ×P = r, where P is the
matrix of transition probabilities of X.
第十一題 Sample space S consists of the permutations x=(x1, x2, ..., xn) of
{1, 2, ... , n} satisfying Σ i * xi < (n^3) / 4.
1≦i≦n
Suppose that for each permutation x of S, the probability of x is
h(x) / Σ h(y) , where h(x) = Σ i^3 * xi.
y∈S 1≦i≦n
Please describe how to estimate the expectation of h(x) over all permutations
x∈S using the Hastings-Metropolis method.
第十二題 在高速公路上統計黃綠紅三種顏色的車輛。每輛黃車後面緊跟著的車輛顏色
的黃綠紅比例為2:1:1，每輛綠車後面緊跟著的車輛顏色的黃綠紅比例為1:2:1，每輛
紅車後面緊跟著的車輛顏色的黃綠紅比例為1:1:2，請估計整條高速公路上這三種顏色
車輛的比例為何？
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