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Unformatted text preview: ' 200 Chapter 6 Describing Probability Ill: Uncountable St and Densities Iteration giyes ,  angst—2.3..
“”‘mm+n tm+n_a””“ Referring to the n = 1 case yields .
(n  I)! (n —1)!(m_1)!_1*<h>r(m> In": 1110:! + 1) (m+ +1521?“ = Tm +n  l)!  Ml"(m+n) We have completed the veriﬁcation that the £3011, n) integtates to 1 and is a legitimate pdf. This
. result holds as well for Ma, b) where a and b are not integers} . EXERCISES ‘E6.1 If the cdf #4 nx<1
. . 1 '
F(x)= ~2~ 1f15x<2.
1 H251
evaluate the pdf'f‘(x)t.
E612 If
' 0 ﬁx; 1
1—3—1 1f —1 5 x < 0_
.ae): 1 . _ _y
E—Zsté 1f 0 5 x < 2
l ifx > 2 _ then evaluate the pdffxcc).
E63 What' 1s the probability that the phase '<l> of the AM catriet received on my cat tadio at  a given time is within 30° of zero phase?
E6.4 If X is random over the interval [ ,1 2], then evaluate P(0_ < X < i).
 E65  Show that the location of the maximum of the Ma, b) density, for a and 19 greater than
1, is at (a 1)/(a + b 2) Is this maximum unique?
E646 If the probability of heads p for a coin is chosen according to 5(2, 2), what is the . probability of a head? _
E6.7 What IS the probability of 2 successes in 4 unlinked repeated trials when the probability p for a success in a single trial' is chosen according to the 5(3, 3) prior?
E6._8 It IS obsetved that, for a particular kind of chip, it is as likely to fail befoxe 5, 000 hours as it is to last longer than 5, 000 hours, 
a. Can you determine the mean chip life? Exercises  ' ' _ 116.9 E6.10
E6.11
116.12
E6.13 E6.14 1316.15 E6.16 E6.l7 E6.18 E619 E620 E621 ' 201 b. What is the probability that a chip will last either less than 1,000 or more than
10,000 hours? If a component that fails without aging has a mean or average lifetime of 5 years, what
is the probability that it will fail before 10 years? The probability is 5 that a phone call will exceed 1 minute in length. What is the
probability that its duration will exceed .5 minute? The waiting time T between successive arrivals of dopant atoms in an implant process is
such that P(T > 1)—  .1.. What 13 the probability description of T? If the probability of a waiting time T between the arrival of successive dopant atoms
exceeding 5 units is. .1, what rs the probability of T > 10? Making reasonable assumptions, if the mean time to failure of a power transistor is 10,000
hours, then what 1s the probability of failure' 1n the ﬁrst 1,000 hours of operatiOn? If the waiting time T to the next photon emission is such that P(T > 2): 0.9 what is
the probability that if we observe this photon source for time 1:, we will see at least one
photon? The waiting time between successive emissions of an electron from a cathode' rs a random
variable T whose average value is l nanosecond (as). A counter for these electrons will
count correctly provided that no two electrons are emitted within less than .05 ns. What
is the probability that the third emitted electron will not be confused with the second emitted electron (that their arrivals are at least .05 ns apart)?
a. If the mean waiting time between the arrivals of successive photons at a detector is l as, what is the pdf for the waiting time T2 between the arrivals of the ﬁrst and
third photons? I
b. What is the cdf for T2?
Photons arrive at a detector so that the mean waiting time between arrivals of successive
photons is 2. a. What rs the pdf of the waiting time T3 to the arrival of the third photon? b. Evaluate the cdt F13 for T3.
c Using your computational resources, determine the median value of T3? For a particular class of computationally difﬁcult problems, it is known that if the algo
rithm is initialized randomly, then the running time R is at least one time unit and the
probability is. 5 that the running time R > 10 time units..What is the probability that the
running time will exceed 1,000 time units? A computer center providing Internet access observes that half of all sessions last for at
least 20 minutes. If the minimum length of a session is one minute, what' 1s the probability of a session length exceeding one hour?
An ftp session of duration L has a minimum time (in appropriate units) of to. We have determined that P(L_ > 2119— 4P (L > 41:0). 9.. Evaluate the probability that the session length will exceed 101:0. b. Sketch the edf F101). _
For a given site, it is known that the probability is 1/2 that an internet session will last
longer than 1'] seconds, It is also known that the shortest possible such session is 1 second.
Specify and sketch the cdf F7 for the duration T in seconds of such sessions. 202 ‘ Chapter 6 Describing Probability_ Ill: Uncountable £2 and Densities E622 The instantaneous amplitude A of a'speech signal is a realvalued random variable that
is described by'the Laplacian density  so) — yea“W. n: R. a. Express [3 1n terms of or. _ ' _
b. What' is the probability of an amplitude exceeding— l. 136.23 If a Speech amplitude A 1s such that P(A > I)“ .— 3, then what rs the PM > 2)?
E624 A speech amplitude S is such that P(S > 1) =P(S > 0). What is the probability description of S?
136.25 Let X denote the difference in intensities of a randomly selected pair of successive pixels in a scanned video image. a. What probability model do you suggest for X? .
b. If it is known that P(X__ < ~5)——  .,1 then what 18 the probability that 0 < X < 5? E636 A speech signal digitizer has quantization levels spanning the range [—«5 + 5] units If the
probability of» a signal amplitude X exceeding this range is e 1° ,what' is the probability model fx for X?  
E627 In designing an AID quantizer for speech recording, we need to know the probabilities of' large amplitudes so as to set the dynamic range of the quantizer correctly. _ a. Specify the pdf 3‘}; for the random speech amplitude A at a given time.
b. Ifwe know thatP([A>1).a* e" ,for what value I is PGAI > 1:) '_—. 9‘61» ~ ~l/400‘I? E628 A neis'y voltage V in a circuit is as likely to be negative as it is to be positive. If
P(V > 1)“  ¢(2), then evaluate P(V < 5). ' E639 If a thermal noise voltage V is such that P(——. 001 < V < .001) ~— 00.6, then estimate
P(l — .001 < V < l + ...001) (Make reasonable approximations.) ' E630 If a noisy current I has mean In and'variance 02, then evaluate P([I i_ < 2) as fully as . you can. .
E631 If the probability of a thermal noise voltage V exceeding 1 av is .,1 what can you say about the pdf fv?
E632 A thermal noise voltage V observed over a bandwidth of 1 GHz across a resistor of 1 megohm is such that the probability is .001 that its magnitude [VI is less than 2
microvolts (2 x 10"6 v). .
a. What Is the temperature T of the resistor?
b What can you say about the probability that [V] will be less than 1 millivolt
(103 v)?
E653 If the probability of a thermal noise voltage V exceeding 1 11V is .49 what can you say about the pdf. fv?
E634 a. A thermal noise source is observed to transfer 1 nano'watt of avze'rage noise power to a load resistance of 1,000 ohms Noting that the variance 0'2 of the voltage is
the average squared voltage, describe the voltage V across this load resistance due
to the thermal noise source. b.. Approximately evaluate HIV} _<_ 101‘). Exercises l I 203 .wu—n....._..____—__——m~______~_.____.m_—._—uu..___—___ E635 A satellite antenna, matched toa SOohm transmission line, is pointed at the Earth and
 a voltage V is measured across a SOohm load in alGHz bandwidth. Neglect other sources (eg, microwaves, etc .) of noise. a. Evaluate the average thermal noisepower P dissipated in the load.
b. Provide an expression for the probability that the magnitude of the thermal noise
voltage V appearing across the load will be less than 2' ”V. E636 The random voltagegenerated by an antenna is modeled by S2 = R and the .N(0, 62)
density or probability law, We are interested in the events .' . Ak={w:lwl>ka}, B={w:w>2o}. Evaluate P(Ak) for k '=1, 2, 3, 4, and P03).
E637 In a standardized test like the SAT, where the outcome is described by .,N'(500 10,000), how likely" 1s it that a student’s score will exceed.700?
E638 Specify the pdf needed to model the following random sources: a. speech amplitudeA, when PUAI, < 1) «— 5 
b.. lifetime L of a lightbulb having an average lifetime oi'1,000 hours;
c. voltage V recorded across a resistor R at temperature T when the voltage is _ . measured across a bandwidth W Hz;
11. duration of. an ftp session when it is four times as probable that it will last as long
as 5 time units as that it will last as long as 10 time units. E639 An AM receiver that' rs not receiving a signal will have a random output 2(t) from its
envelope detector that rs due to channel and front end noise. It' 1s often the case that Z at
' a ﬁxed time has what' rs called a Rayleigh distribution and it can be described as follows. :[0 cc). . _ _ ,
P(Aj_ — [ azzei'mﬂzﬂdz.
A a. Verify that this speciﬁcatiOn satisﬁes the basic probability axioms K1, K2, and K3.
b. Evaluate the probabilities of the following events:  ”40 — {Z}, Al=[01z)s AZ "‘ (z/tx, 00)" [£6.40 Provide an expression for the probability, when no signal is present, that the output of an AM envelope detector will exceed a value of v.
E6.41 In cellular communications, it is common to model the received power P due to fading from a base station by the [agnormal distribution, _ ____1_ e3; '
.fptx)— ”ﬁe U(x) Verify that this' re a pdf.
E642 Assume that the maximum power _P (in gigawatts) demanded daily from a large power
plant rs a random quantity, ﬂuctuating from day to day, with probability P(A) of lying .111 the set A given by P(A) = j craze—2" dp..
Amman)  ' 204 Chapter 6 Describing Probability ill: Uncountable $2 and Densities a. Find the value of c needed to ensure that P satisﬁes the Kolmogorov axioms. Recall that
00
f x"e”xdx = k!..
0 b If the plant has a maximum capacity of 3 gigawatts, then what' is the probability
that on a given day demand wiﬁ exceed capabity? ...
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 Spring '05
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