Unformatted text preview: 402 Chapter 7 Sums of Random Variables and Long—Term Averages Problems 403 3. S M Ross. Introdurtion. to Probability Models Academic P1ess Newy
I 4. J. A. Cadzow, Foundations of Digital Signal Processing and Data
Macmillan, New York, 1987. . _ p . 5. P. L. Meyer, Introductory Probability and Statistical Applications, 2nd 9 son—Wesley, Reading, Mass, 1970.  ﬂ 6. J. W. Cooley. P. Lewis, and P. D. Welch, “The Fast Four1er Transform and ,_ cations,” IEEE Transactions on Education, V01. 12, pp. 27—34, March 1969 7. H. J. Larson and B. O. Shubert, Probability Models in Engineering Science 1 Wiley, New York 1979.
8. 11. Stark and J. W. Woods Probability and Random Processes withAppzle Signal Processing. 3d ed, P1ent1ce Hall, Upper Saddle River, NJ, 2002. (c) Show that T2 is a Rayleigh random variable. (d) Find the pdf for ’15. The random variable T3 is used to model the speed of molecules
in a gas. T3 is said to have the Maxwell distribution. . Let X and Y be independent exponential random variables with parameters 2 and 10, re
spectively. Let Z = X + Y. (3) Find the characteristic function of Z. (b) Find the pdf of Z from the characteristic function found in part a. 8. Let Z : 3X 1. 71’, where X and Y are independent random variables.
(:1) Find the characteristic function of Z. (b) Find the mean and variance of Z by taking derivatives of the characteristic function
found in part a. .9. Let M” be the sample mean of n iid random variables X1. Find the characteristic function of M. in terms of the characteristic function of the X 1’s
PROBLEMS ' l 210 “10 number X 0f rafﬂe Wlnners in classroom j is a binomial random variable with para—
Section 7.1: Sums of Random Variab es meter n andp. Suppose that the school has K classrooms. Find the pmf of the total num—
ber of rafﬂe Winners in the school assuming the X,’ s are independent random variables The number of packet anivals X at port i in a route1 is a Poisson random variable with mean ai. Given that the router has k ports, find the pmf for the total number of packet ar—
rivals at the router. Assume that the Xi’s are independent random variables. 7_1_ Let Z = X + Y + 23where X , Y, and Z are zero~mean,unitvarianee random
with COV(X. y) : 1/2, and COV(Y, Z) = —1/4 and COV(X, Z) 2 1/2. (3) Find the mean and variance of Z.
(b) Repeat part a assuming X. Y , and Z are uncorrelated random variables. . . . . , Let X X a . . . be a sequence of independent integervalued random variables let N be
. nd With covar ' 1’ a” 7
7'2‘ Let X1" ‘ ' ’ X” be random var1ables Wlth the same mean a lance an integer—valued random variable independent of the X 1" and let
2  . _  1N7
o3 1fi~]. 5:2Xk.
COV(X.. X.) 2 p0“ if li — ii : i=1
0 otherwise.1(a) Find the mean and variance of S.
(b) Show that
where pl < 1. Findthe mean andva1iance of S” — X1 + “l X”. gr ) = E( S) : G (G ( ))
7.3. Let X 1 , .. ,,,X be random variables with the same mean and with covarian A Z Z a x Z 7 where Gy<Z> is the generating function of each of the Xk’s. Let the number of smashed— up cars arriving at a body shop in a week be a Poisson ran dom variable with mean L. Each job repair costs X dollars, the X s a1e iid random vari
ables that are equallv likelv to be $500 or $1000 (3) Find the mean and variance of the total revenue R arriving in a week.
(b) Find the ow) : E; ZR]. Let the number of widgets tested in an assembly line in 1 hour be a binomial random COV(X,, x',)— 1 a p” l where I p I 1. Find the mean and variance of Sn — X1+ ~ ~  + X“. 7. 4. Let X and Y be independent Cauchy random variables with parameters 1 an
t1vely. LetZ # X + Y. (:1) Find the characteristic function of Z.
([1) Find the pdf of Z from the characteristic function found in part a , variable With Parameters n =600 and p. Suppose that the probability that a widget is
7 5 Let $162+ ' l I + Xk’ where the X S a1ef1nd:1pend:1111t midi”: 222:1]? 1au1tyis a. Let S be the numbei of widgets that me found faulty 111 a 1 —hour period.
Chi— —square random variable with+ n degrees of ree om ow a ,  (3) Find the mean and variance of S.
dom variable with n _ m + +Ilk deglees of freedom. (J ([1) Find GS(Z) : ﬁkSJ
7.(. Let Sn = X i + X2, where the X s a1eiid zero— —mean unit variance au dom variables. (a) Show that S” is a chi—square random variable with n degrees of freedo
Example 4.34. (1)) Use the methods of Section 4.5 to find the pdf of ection 7.2: The Sample Mean and the Laws of Large Numbers Suppose that the number of particle emissions by a radioactive 1n ass in 1 seconds is a Pois— son random variable with mean At. Use the C hebyshcv inequality to obtain a bound for
the probability that lN(t t)/t e AI exceeds a. 16 Suppose that 20% of vote1s are in favor of certain legislation A la ge number n of voters Tn = V X1 l ' ' + X14" are polled and a relative f1equency estimate f4(n) for the above p1 opo1t1on is obtained ...
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