1 ECE 340 Homework #5 Solutions
ECE 340: PROBABILISTIC METHODS IN ENGINEERING
SOLUTIONS TO HOMEWORK #5
3.35
a) By the definition of conditional probability,
P({X=k}|{X >0}) = !( !!!}β©{!!! )! !!!
= ! !!! ! !!!
= ! !!!!"/!"
=
8161516
=815, π = 1
7161516
= 7/15, π = 2
Note: we used the fact that {X=k} β {X>0} for k=1, 2.
b) P({X=k} | {Nm = 1}) = !( !!!}β©{ !!!! )! !!!!
= !( !!!}β©{ !!!! )!/!
=
14
12 + 1
212
1/2=34, π = 1
14
12
12
=14 , π = 2
3.44 a) S={1,2,3,4,5} A={ΞΆ > 3}. Llet us assume that all outcomes are equally likely. Now,
P{IA=0} = P{Ac}=P{1,2,3}=3/5; P{IA=1} = = P{A}=P{4,5}= 2/5; E[IA] = 0 x 3/5 + 1 x 2/5 = 2/5.
b) S=[0,1] A={0.3 < ΞΆ < 0.7} Again let us assume that we have uniform outcomes; that is, if E is an event, the P(A)=length(E). P{IA=0} = P{Ac}=0.3+0.3 = 0.6 P{IA=1} = P{A} = 0.4 E[IA] = 0 x 0.6 + 1 x 0.4 = 0.4
2 ECE 340 Homework #5 Solutions
c) S={ΞΆ = (x,y): 0 < x < 1, 0 < y < 1} and A = { ΞΆ = (x,y): 0.25 < x+y < 1.25}
IA(ΞΆ )= 0 when the pair (x,y) is not in the indicated area. IB(ΞΆ )= 0 when the pair (x,y) is in the indicated area. The probabilities can be calculated by computing the areas
P{IA=0} = !.!"!
!+ !!!.!" !
!= !
!"= 0.3125
P{IA=1} = 1- P{IA=0} = 1 β !
!"= !!
!"= 0.6875
E[IA] = 0 x 0.3125 + 1 x 0.6875 = 0.6875
d) S=(-β , β ) and A={ΞΆ >a} P{IA=1} = P(A) P{IA=0} = P(Ac)=1-P(A) E[IA] = 1 x P(A) = P(A)
3.45 A and B are events from a random experiment with sample space S. a) Show that, IS = 1
From the definition of the indicator function we have that,
πΌ!(π) = 0 ππ π πππ‘ ππ π1 ππ π ππ π
However, by the definition of sample space, all the points ΞΆ have to be in S, so IS=1 for any ΞΆβΞ©. Likewise, we show that, IΓ = 0
1
1
0.25
0.25 x
y
A 0.25
0.25
3 ECE 340 Homework #5 Solutions
From the definition of the indicator function we have that,
πΌΓ(π) = 0 ππ π πππ‘ ππ Γ1 ππ π ππ Γ
but by the definition of the null event Γ, contains no outcomes and hence never occurs, so IΓ=0 for any ΞΆβΞ©.
b) IAβ©B = IA IB β’ π°π¨β©π© π» = π if π β π΄ πππ π β π΅, which corresponds to IA=0 and IB=0; hence, IA IB =0
or if π π π΄ πππ π β π΅, which corresponds to IA=1 and IB=0; hence, IA IB =0 or if π β π΄ πππ π β π΅, which corresponds to IA=0 and IB=1; hence, IA IB =0.
β’ π°π¨β©π© π» = π if π π π΄ πππ π π π΅, which is corresponds to IA=1 and IB=1; hence, IA IB =1. To show that, IAUB = IA + IB - IAβ©B notice that β’ π°π¨πΌπ© π» = π only if π β π΄ πππ π β π΅, which is corresponds to IA=0 and IB=0, and IA
+ IB - IAUB = 0 β’ π°π¨πΌπ© π» = π if π π π΄ πππ π π π΅, which in this case makes to IA=1 and IB=1 and IAUB=1,
and IA + IB - IAUB = 1 or if π π π΄ πππ π β π΅, which corresponds to IA=1 and IB=0 and IAUB=0, so IA + IB - IAUB = 1 or if π β π΄ πππ π β π΅, which is equivalent to IA=0 and IB=1 and IAUB=0, so IA + IB - IAUB = 1
c) The expected values are
E[IS]= 0pI(0) + 1pI(1) = pI(1) = 1 E[IΓ]= pI(1) = 0 E[IAβ©B]= pI(1) = P(A β© B) E[IAUB]= pI(1) = P(A U B) = P(A) + P(B) β P(A β© B)
4 ECE 340 Homework #5 Solutions
3.48 The pmf of a binomial random variable is given as
ππΏ π = π· πΏ = π =ππππ(π β π)π!π, π = π,π,β¦ ,π
The pmfs of various values of n and p are given in the following figures for n = 4 and p = 0.1
for n = 5 and p = 0.1
for n = 4 and p = 0.5
0
0.2
0.4
0.6
0.8
0 1 2 3 4 5
Series1
0
0.2
0.4
0.6
0.8
0 1 2 3 4 5 6
Series1
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5
Series1
5 ECE 340 Homework #5 Solutions
for n = 5 and p = 0.5
for n = 4 and p = 0.9
for n = 5 and p = 0.9
3.49 a) Let IK denote the outcome of the kth Brenoulli trial. The probability that the single
event occurred in the kth trial is
π πΌ! = 1} {π = 1 =π πΌ! = 1 } β© { πΌ! = 0 πππ πππ π β π
π π = 1
=π 0, 0,β¦ ,1, 0, . .0
π π = 1
0 0.1 0.2 0.3 0.4
0 1 2 3 4 5 6
Series1
0
0.2
0.4
0.6
0.8
0 1 2 3 4 5
Series1
0
0.2
0.4
0.6
0.8
0 1 2 3 4 5 6
Series1
6 ECE 340 Homework #5 Solutions
=π 1 β π !!!
π1 π 1 β π !!!
=1π
Thus, the single event is equally likely to have occurred in any of the n trials. b) The probability that the two successes occurred in trials j and k is
π( πΌ! = 1, πΌ! = 1 π = 2}) =π({πΌ! = 1, πΌ! = 1} β© { πΌ! = 0 πππ πππ π β π, π})
π{π = 2}
=π! 1 β π !!!
π2 π! 1 β π !!!
=1π2
c) If X=k, then the locations of the successes are randomly selected from the ππ
possible permutations of possible locations. The concept of completely at random can be interpreted as the concept of βuniformly distributedβ as can be seen from parts a) and b).
3.52 p=0.01 N=number of error-free characters until the first error.
a) P({N=k})=(1-p)kp k=0,1,2,β¦ b) πΈ[π] = π 1 β π !π = 1 β π π π 1 β π !!!!
!!!! !!!
= 1 β π π1
1 β 1 β π! =
1 β ππ
ππ¦ πΈπ. (3.14)
c) 0.99 = π π > π! = 1 β π !π = π 1 β π !!!! 1 β π !!
!!!!!!!!
= 1 β π !""! β π = 1 β 0.99!
!""! = 1.004π₯10!!
3.56 Letβs denote the random variable X as the cost of repair of the audio player in one year period, and denote c as the charge for the one year warranty. The pmf of r.v. X can be found as
π π = π₯ = !"(!/!")
!!"
!!" (1 β !
!")!"!
!!" for π₯ = 0, 20, 40, 60,β¦ , 240.
Thus,
π π = 0 =12020
112
!!"(1 β
112)!"!
!!" = 0.351995628014137
π π = 20 = 0.383995230560877
7 ECE 340 Homework #5 Solutions
π π = 40 = 0.191997615280438 π π = 60 = 0.058181095539527 π π = 80 = 0.011900678633085 π π = 100 = 0.001731007801176 π π = 120 = 0.000183591736488 π π = 140 = 0.000014305849596 π π = 160 = 0.000000812832363 π π = 180 = 0.000000032841712 π π = 200 = 0.000000000895683 π π = 220 = 0.000000000014805 π π = 240 = 0.000000000000112
If now we want the probability of losing money on a player is 1% or less, we want π[π > π] β€ 0.01. And the question now is βwhat value should the c take such that the previous inequality is satisfied?β Note that P[X>c] is monotonically decreasing as c increases and P[X>60] = 0.013830430605021 > 0.01; and P[X>c] = 0.013830430605021 > 0.01 for any c in the interval of (60,80); P[X>80] = 0.001929751971936 <= 0.01; P[X>c] <= P[X>80] = 0.001929751971936 <= 0.01 for any c >= 80. So the manufacturer should charge at least 80 dollars to make sure the probability of losing money on a player is 1% or less (excluding the initial cost of $50.).
The average cost (of repair) per player is πΈ π = π₯! π π = π₯ = 20 dollars. This expectation can be calculated by the following matlab code:
for x = 0:20:240 cost((x+20)/20) = x*(nchoosek(12,x/20))*((1/12)^(x/20))*((11/12)^(12-(x/20))); end >> sum(cost) ans = 19.999999999999996 >>
3.59 (Please also see Example 3.30 on page 122)
a) We know that the average is 6000 requests per minute, which is equivalent to 0.1 requests per millisecond, which corresponds to having Ξ± = Ξ»t = 0.1t in the Poisson distribution (where t is in milliseconds). The probability of having no requests in a 100-ms period is
P{N=0} = (!")!
!!π!!" = π!!" = π!!.!β!"" = π!!" = 45.399 β 10!!
8 ECE 340 Homework #5 Solutions
b) The probability that there are between 5 and 10 requests in a 100-ms period
π{5 β€ π β€ 10} =ππ‘ !
π!π!!" =
!"
!!!
0.1Γ100 !
π!π!!.!Γ!"" =
!"
!!!
0.5538
3.62 Solution: The Poisson r.v. has the following pmf:
π π = π = π! π = πΌ!
π!π!! , πππ π = 0, 1, 2,β¦.
π!π!!!
=
πΌ!π! π
!!
πΌ!!!π β 1 ! π
!!=πΌπ
πΌπ πΌ < 1 π‘βππ π!π!!!
=πΌπ< 1 πππ π β₯ 1
β΄ π! πππππππ ππ ππ π πππππππ ππ ππππ 0
β΄ π! ππ‘π‘ππππ ππ‘π πππ₯πππ’π ππ‘ π = 0
πΌπ πΌ > 1 π‘βππ (Note that we denote by [x] the largest integer that is smaller than or equal
to x.)
If 0 β€ π β€ πΌ < πΌ, π‘βππ !!!!!!
= !!> 1
Hence, pk increases from k=0 to k=[Ξ±]
If πΌ β€ πΌ β€ π, !!!!!!
= !!< 1
and pk decreases as k increases beyond [Ξ±]
β΄ π! ππ‘π‘ππππ ππ‘π πππ₯πππ’π ππ‘ π!"# = [πΌ]
πΌπ πΌ = πΌ π‘βππ πππ π = [πΌ]
π!π!!!
= 1, and hence, π!!"# = π!!"#!!
3.68 Solution:
πΈ[π] = ππ!
!!!
π = π = π1πΏ
!
!!!
=1πΏ
π!
!!!
=πΏ(πΏ + 1)2πΏ
=πΏ + 12
π!! = πΈ[π!] β πΈ[π]! = π!1πΏ
!
!!!
βπΏ + 12
!=πΏ πΏ + 1 2πΏ + 1
6πΏβ
πΏ + 1 !
4=πΏ! β 112
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