課程名稱︰機率與統計
課程性質︰選修
課程教師︰黃乾綱
開課學院：
開課系所︰工程科學及海洋工程學研究所/系
考試日期（年月日）︰2013/1/10
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 :
1.[CH4](30%)Random Variables X and Y have joint PMF.
Px,y(x,y)= cxy x=1,2,4; y=1,3
0 otherwise
And the function W=X-Y
a. (3%)What is the value of the constant c?
b. (4%)The marginal PMFs Px(x) and Py(y).
c. (4%)The expected value E[X] and E[Y].
d. (4%)The standard deviations sigmax and sigmay.
e. (3%)The probability mass function Pw(w).
f. (2%)The expected value E[W].
g. (3%)What is pP[W>0]?
h. (3%)The correlation, E[XY]
i. (2%)The covariance, Cov[X,Y]
j. (2%)The correlation coefficient of X,Y
2.[CH4](15%)Random Variables X and Y have joint PDF
fx,y(x,y)= 1/2 -1 <= x <= y <= 1
0 otherwise
a. What is fy(y)?
b. What is fx|y(x|y)?
c. What is E[X|Y=y]?
3.[CH5](15%)The 4-dimensional random vector X has PDF:
fx,y(x,y)= 1 0 <= xi <= 1, i=1,2,3,4
0 otherwise.
a. (3%)Are the four components of X independent random variables?
b. (4%)Find the expected value vector E[X]
c. (4%)Find the correlation matrix Rx
d. (4%)Find the covariance matrix Cx
4.[CH6](15%)Let K1, K2, ... denote a sequence of iid Bernoulli(p)
random variables.
1-p x=0
Px(x)= p x=1
o otherwise
Let M=K1+K2+ ... +Kn
a. Find the MGF K(s)
b. Find the MGF M(s)
c. Use the MGF M(s) to find the expected value E[M] and variance Var[M]
5.[CH6](25%) The waiting time W for accessing on record from a computer
database is a random cariable uniformly distributed between 0 and 10
milliseconds. The read time R(for moving the information from the disk
to main memory) is 3 milliseconds. The random variable X milliseconds
is the total access time (waiting time + read time, that is, X=W+R)
to get one block of information from the disk. Before performing a
certain task, the computer must access 12 different blocks of information
from the disk. (Access times for different blocks are independent of
one another) The total access time for all the information is a
random varaible A milliseconds.
(That is, A=X1+ ... + X12, is the sum of 12 iid random variables)
a. What is E[X] the expected value of the access time?
b. What is Var[X], the variance of the access time?
c. What is E[A] the expected value of the total access time?
d. What id sigmaA, the standard deviation of the total access time?
e. Use the central limit theorem to estimate P[A > 116ms],
the probability that the total access time exceeds 116ms.
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