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. 1-1. 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 2-10 “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,unit-varianee 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 integer-valued 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
o-3 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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