Serie de Tutoriales en Español Sobre Unreal Engine 4
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ECB3133 Probability and Random Process (May 2014)
Tutorial ee! 4 1" #or any t$ree t$ree e%ents e%ents &' B and and C' use basic robability robability las las to s$o s$o t$at* t$at* P(& ⋃ B ⋃ C) + P(&) , P(B) , P(C) - P(& ⋂B) - P(&⋂C) - P(B⋂C) , P(&⋂B⋂C)" 2" T$e dia.ram dia.ram belo reres reresents ents a communica communication tion netor! netor! $ere $ere t$e source source S communica communicates tes it$ t$e recei%er R" /et & reresent t$e e%ent t$at lin! a ails and B reresent t$e e%ent lin! b ails and so on" rite an eression in t$e al.ebra o e%ent or t$e e%ent # + S ails to communicate it$ R"
a
b c
S d
R e
&ssume t$at t$e lin! ailures are indeendent and t$at eac$ lin! can ail it$ robability 0"1" Comute t$e robability o t$e e%ent #" 3" n a di.ital di.ital control control system' system' t$e control control command command is reresente reresented d by a set o our $eadecim $eadecimal al c$aracters" a) $at is is t$e total total number number o control control commands commands t$at are ossibl ossible5 e5 b) eac$ control co ntrol command must $a%e a uni6ue rei' i"e' startin. rom let to ri.$t no $e c$aracter must be reeated' $o many control commands are ossible5 4" & si.na si.nall $as been been saml samled ed'' 6uan 6uanti ti7ed 7ed (8 le%el le%els) s) and encod encoded ed into into 3 bits bits"" T$es T$esee bits bits are se6uentially transmitted o%er a ire" a) 9ra t$e samle samle sace correso corresondin. ndin. to t$e recei%ed recei%ed bits" bits" &ssume &ssume t$at t$at no errors occur occur durin. t$e transmission" b) 9etermine t$e robability t$at t$e recei%ed encoded samle $as a %alue .reater t$an :" c) $at $at is t$e rob robab abil ilit ity y t$at t$at t$e t$e %alu %aluee o t$e t$e rece recei% i%ed ed sam samle le is bet betee een n 3 and and ; (inclusi%e)5 :" n eac$ lot o 100 items' items' 2 items items are tested' tested' and t$e t$e lot is re
ECB3133 Probability and Random Process (May 2014) robability by p( N )" #ind t$e %alue o N t$at maimi7es t$is robability" >int* Comare t$is ratio p( N ) ? p( N -1) to unity" @" & ternary communication c$annel is s$on in #i." 1" Suose t$at t$e inut symbols 0' 1 and 2 occur it$ robability A' and ' resecti%ely" a) #ind t$e robabilities o t$e outut symbols" b) Suose t$at a 1 is obser%ed as an outut" $at is t$e robability t$at t$e inut as 05 15 25
n u 0 t 1
=ut ut 0
1ε
1ε ε
ε
1
1ε
2
2 Figure 1
8" T$ree tyes o messa.es arri%e at a messa.e center" 10 o t$e messa.es are $i.$ riority' 40 are normal riority and :0 ercent are lo riority" a) #ind t$e robability t$at ! out o D messa.es are not $i.$ riority" b) Suose t$at messa.es arri%e one at a time" #ind t$e robability t$at ! messa.es are recei%ed beore a $i.$-riority messa.e arri%es" c) #ind t$e robability t$at out o 20 messa.es' : are $i.$ riority' 10 are normal riority and : are lo riority" " n a certain comuter' t$e robability o a memory ailure is 0"01' $ile t$e robability o a $ard dis! ailure is 0"02" t$e robability t$at t$e memory and $ard dis! ail simultaneously is 0"0014' t$en a) &re memory ailures and $ard dis! ailures indeendent e%ents5 b) $at is t$e robability o a memory ailure' .i%en a $ard dis! ailure5 c) Reeat (a) and (b) i t$e robability o a memory ailure is 0"02' t$e robability o $ard dis! ailure is 0"01: and t$e robability t$at bot$ ail simultaneously is 0"0003" 10" & bo contains 1; inte.rated circuits (Cs) o $ic$ 3 are !non to be deecti%e" a) Fou are as!ed to randomly select to Cs rom t$e bo" $at is t$e robability t$at at least one C is .ood5
ECB3133 Probability and Random Process (May 2014) b) you randomly select one C rom t$e bo' $at is t$e robability t$at it is .ood5
&nsers to Tutorial ee! 4 1" Gse basic robability la on union o sets and distributi%e la" # + S ails to communicate it$ R + (& ⋃ B) ⋂ C ⋂ (9 ⋃ E)
2"
P (#) + 0"003;1 a) Total ossible number o commands + 1;4
3"
16
b) Dumber o commands it$ no c$aracters reeated +
P4
43680
=
4" a) S + H000'001'011'010'110' 111' 101' 100I b) P (J K :) + 1?4 c) P (3 L J L ;) + A :" a) P (a lot is acceted it$ : deecti%e items) + 0"02 P (a lot is acceted it$ 10 deecti%e items) + 0"80 b) P (a lot is acceted it$ : deecti%e items) + 0"4 P (a lot is acceted it$ 10 deecti%e items) + 0"@4 ;" N + 40 @" a) P (=0) + A P (=1) + (1,N) P (=2) + (2-N) 2ε
b)
8" a)
( I |O ) = (1 + ε ) 0
1
()
P N ( k )=
N k
k
( 1 −ε )
( | )= (1+ ε )
P I 1 O 1
( | )= 0
P I 2 O1
N − k
0.9 0.1
b) P (! messa.es recei%ed) + 0"k c) P O(:>' 10D' :/) + 1":2; 10-3 " a) E%ents M and > are not indeendent b) P (M Q >) + 0"0@ c) M and > are indeendent e%ents' P (M Q >) + 0"02 + P (M) 10" a) P (at least 1 C is .ood) + 0"@:
ECB3133 Probability and Random Process (May 2014) b) P (1 C is .ood) + 0"812:
