Download 等候理論HW#1

等候理論 HW#1
Due:
1. A certain car manufacture has factories A, B, and C producing, respectively, 35
percent, 15 percent, and 50 percent, of the total output. The probabilities of a
car NOT being defective is 0.70 if it was produced at factory A, 0.95 if it was
produced at factory B and 0.85 if it was produced at factory C. A customer buys
a defective car.
What is the probability that it came from factory C?
2. Find the value of the constant k such that
f(x)=kx2(1-x3)
0x1
=0
otherwise,
is a proper density function of a continuous random variable.
3. Consider discrete random variables X and Y with the joint pmf as shown below:
Y=-1
Y=0
Y=1
X=-2
1/16
1/16
1/16
X=-1
1/8
1/16
1/8
X=1
1/8
1/16
1/8
X=2
1/16
1/16
1/16
Are X and Y independent? Why?
4. Assume that the probability of successful transmission of a single bit over a
binary communication channel is p. We desire to transmit a four-bit word over
the channel. Scheme 1 is to transmit the word directly. To increase the
probability of successful word transmission, we may use a new code (four data
bits + three check bits). So we now transmit 7 bits where before we only
transmitted 4 bits. Assume that under scheme 2, the new code is known to be
able to correct only single-bit error, i.e., if two or more bits of the code word are
in error then the word is not correctly received.
(a) What is the probability of successful word transmission under each of the
two schemes.
(b) For what value of p will the use of the new code increase the probability of
5.
successful transmission of a word?
Consider the case statement:
case N of
1: S1;
2: S2;
…….
n: Sn;
end;
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Let the mutually independent random variables X1, …, Xn respectively, denote the
execution times of the statement groups S1, …, Sn. Let P[N=i] = pi, I=1, 2, …,
n, andni=1 pi = 1. Let X be a random variable equal to the execution time of
the case statement (not including the time to select the correct statement)
Compute E[X] and Var[X] in terms of the means and variances of X1, …, Xn, and pi.
6. Suppose X and Y are independent random variables uniformly distributed over {1,
2, …, n}.
(a) Find P(XY)
(b) Find P(X=Y)
(c) Find the pmf of Z1=X+Y.
(d) Find the pmf of Z2=max{X,Y}
(e) Find the pmf of Z3=min{X,Y}
7. Consider a computer system, for which, when the system fails, the repair time is a
random variable T. The number of jobs arriving during the repair interval, N, is
a random variable with Poisson distribution with parameter t when T=t.
Let fT(t) be the pdf for T,
FT(t) be the distribution function for T,
F*(s) be the Laplace transform of fT(t).
Express the PGF of N in terms of F*(s) and .
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