課程名稱︰高等統計推論
課程性質︰選修∕數研所統計組必修
課程教師︰陳宏 教授
開課系所︰數學系
考試時間︰2005/11/23 13:20～15:30
試題：
Instructions. You have 110 minutes to complete the examination. This is a
closed-book examination.
1. (40％) Assume that U is a continuous uniform random variable taking values
j-1 j ∞
0 and 1. Let X be j if Σp_i≦U≦ Σp_i. Here p_j＞0 and Σp_j＝1,
0 i=0 i=0 j=0
Σp_j＝0.
j=0
(a) (30％) Determine the probability distribution of X. -2
(b) (10％) Prove that the mean of X does not exist when p_j＝j ∕c. Here
∞ -2
c＝ Σ j .
j=0
2
2. (50％) The probability density function of X is h(x)‧exp(-x∕2), where
_____ 2
x 屬於Ｒ and h(x)＋h(-x)＝√2∕π. Let Y＝X .
(a) (30％) Determine the probability density function of Y.
(b) (20％) Determine E(Y).
In this problem, the following fact can be used. For any α,β＞0,
α ∞ α-1
Γ(α)β ＝∫ x exp(-x∕β)dx.
0
3. (50％) Suppose X and Y have the following joint probability density
function
f(x,y)＝(2－x－y)‧I_(0,1) ×(0,1)(x,y).
2
(a) (30％) Find P(X ＜Y＜X).
(b) (20％) Describe the marginal distribution of X.
X
4. (40％) Suppose that X is a nonnegative random variable with E(t )＝
exp(t－1), t＞0. Based on this information, what is the best upper bound
we can have for P(X≧2)?
5. (60％) In a digtal communication system, 0's and 1's (bits) are transmitted
as signal waveforms. Suppose that any bit to be sent is a 0 with
probability 0.5 and a 1 with probability 0.5, independently from bit to
2
bit. If a 0 is transmitted it is received as a waveform X that is N(-1,σ )
2
and if a 1 is transmitted it is received as a waveform X that is N(1,σ ).
For each bit sent the receiver must decide if a 0 or 1 was sent. If the
receiver incorrectly decodes a transmitted waveform we say a bit error has
occurred.
(a) (40％) Consider the decision rule which decodes a 1 if the received
waveform X≧0 and decodes a 0 if X＜0. If π are decoded in error in
terms of the distribution function of a standard normal random variable
Φ(.).
(b) (20％) What is the largest upper bound on σ to ensure π≦0.05?
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