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LOGO Chapter 2 Discrete Radom Variables(R.V) Part1 iugaza2010.blogspot.com
2.1 Radom Variables A radom variable over a sample space is a fuctio that maps every sample poit (i.e. outcome) to a real umber. The mai purpose of usig a radom variable is so that we ca defie certai probability fuctios that make it easy to compute the probabilities of various evets.
R.V Discrete Cotiuous Ex: Roll a fair coi oce: Outcomes={H,T } X 1 0 if if head Tail 4
Ex: Toss a coi twice. Let Y deote the umber of heads. 5
P Y (y) 1/4 1/2 1/ 4 y 0 y 1 y 2 he probability mass fuctio of a fair die 6
The R.V N has PMF P N ( ) c(1/2) 0 0,1,2 else (a) Fid the value of the costat c? P( N 0) P( N 1) P( N 2) 1 c *1 c *(1/ 2) c *(1/ 4) 1 c 4 / 7 (a) N 1] 4 4 1 N 1] N 0] N 1] ( * ) 7 7 2 6 7
1/8 x 0 3/8 x 1 P X ( x) 3/8 x 2 1/8 x 3 0 else Fid : X=0], X<3] X 0] 1/8 X 3] X 0] X 1] X 2] 7/8 8
The R.V N has PMF P V ( v) cv 0 2 v 1,2,3,4 else (a) Fid the value of the costat c? P( V 1) P( V 2) P( V 3) P( V 4) 1 c 4c 9c 16c 1 c 1/ 30 (a) VЄ{U 2 u=1,2,3,4,.] 1 1 V 1] V 4] ( *1) ( *16) 30 30 17 30 9
(c) Fid the probability that V is eve? 1 1 P( V 2) P( V 4) ( *4) ( *16) 30 30 20 30 10
I basketball: whe a player is fouled the player is awarded Oe ad Oe oe free throw ad if it goes i the player is awarded aother oe. Y Fid PMF of Y, the umber of poit scored i 1 ad 1 The throw goes i with probability p {0,1,2} g1b 2 P Y y,,, 1,, b1 y g1g 2 0 y 1- p y 0 p(1- p) y 1 ( y) 2 p y 2 0 else 2 11
You are maager of a ticket agecy that sell cocert tickets. You assume that people will call three time i a attempt to buy tickets ad the give up. You wat to make sure that you are able to serve at least 95% of the people who wat tickets. Let P be the probability that the caller gets through to your ticket agecy. What is the mi value of P ecessary to meet your goal? 12
prob. that the caller fails 1-0.95 F] (1- p) 3 p 0.63 (1- p)(1- p)(1- p) (1- p) 0.05 3 0.05 13
2.3 Families of discrete R.V [1] Beroulli P X ( x) p 1- p 0 x 0 x 1 else Ex: test oe circuit (rejected=0.2 or acceptable=0.8) R.V X=# of rejected 0.8 x 0 P X (x) 0.2 x 1 0 else
2.3 Families of discrete R.V [2] Geometric R.V P X ( x) p(1- p) 0 x-1 x 1,2,3,... else Ex: test oe circuit (rejected=(1-p) or acceptable=p) R.V X=# of test util get the first acceptable
[3] Biomial R.V P X ( x) p x (1 p) x, x x 0,..., Ex: test 10 circuits (rejected=0.2 or acceptable=0.8) R.V X=# of rejected i the 10 tests P X ( x) 10 x (0.2) x 10 (0.8) x
[4] Pascal R.V k Lk L k 1 1 p (1 p) Ex: test circuits util you fid 5 rejected,after you doe 105 tests the 5 rejected is foud. (rejected=0.2 or acceptable=0.8) Prob.of A] Prob.of B] all excpt the fial 1051 5 1 oe 51 (1051) (51) (0.2) (0.8) from biomial p the fial 0.2 oe
Prob.of all A] B] 1051 (0.2) 5 1 excpt the fial 1051 (0.2) 5 1 5 (0.8) (1055) oe 51 (0.8) (1051) (51) *(0.2) Pascal (k-1) rejectsi (L-1)attempts,rejectedo attempt L] 18
P X [5] Discrete uiform R.V 1 ( x) l k 0 x k, k 1,..., 1 else l Ex: roll die {1(k),2,3,4,5,6(L)} P X (x) 6 0 1 11 1 x 6 else 1,2,...,6
[6] Poisso R.V P X ( x) T 0 x - e x! : averagerate x 0,1,2,... else
The probability that the message will be received is p. The message is received at least oce, a system trasmits the message times. (a) What is the PMF of K: the umber of times the receiver receives the same message? k k P K ( k ) p (1 p),, k 0,..., /// biomial k (b) Assume p=0.8, What is the mi value of that produces a probability of 0.95 of receivig the message at least oce? at least Oe] 1 zero] P at 0.2 K (0) l0.2 least Oe] 1.86 0 0.05 l0.05 2 p 0 (1 1 p) 0 0.2 0.2 0.95 21
A child throws a Frisbee, the child s dog catches it with probability p, whe the dog catches the Frisbee it ru away. The child cotiues to throw Frisbee util the dog catches it. Let X deote the umber of times the Frisbee is throw more 1{ x 1{0.2*0.8 (a) What is PMF of X? X P X 0.41 is a geometricr.v ( x) 4 times] 4] 3 p(1 x x 3] 0.2*0.8 p) 5] 2 x1 (b) Assume p=0.2, What is the probability that the child will throw the Frisbee more tha four times? x x, x 2] 0.2*0.8 6] 1 x 1,2,3,... x 1]} 0.2*0.8 7]... 0 } 22
Whe a two-way pagig system trasmits a message, the probability that the message is received by the pager it is set to is p. Whe the pager receives the message, it trasmits a ackowledge sigal (ACK) to the pagig system. If the pagig system does ot receive the ACK, it seds the message agai. (a) What is PMF of N:the umber of times the system seds the same message? N P if N N is a ( ) p 3] (1 geometric R.V p(1 p) p p) 1] (1 1 6 so 5 fails ad 1success (b) The pagig compay wats to limit umber of times it has to sed the same message. It has a goal of N 3] 0.95, what is the mi value of p ecessary to achieve the goal? p) 2, p 2] p 1,2,3,... p p 2 3] p 2 p 2 p 3 23
p 3 3 p 2 3 p 0.95 p 3 3p 2 solvefor p p 0.6315 3p 0.95 0 p P 1,2 3 1.184 0.6315 j0.319(otaccept) 24
The umber of buses that arrive at a bus stop i T miutes is a Poisso R.V B with expected value T/5 (a) What is the PMF of B,the umber of buses that arrive i T miutes? for Poisso R.V \ \ 1/ 5 ( T P B( b) 0 T / 5) : averagerate b -(T/5) e b! else expected value 0,1,2,... (b) What is the probability that i a 2 miute iterval,three buses will arrive? T P B 2, ( b b 3) 3 (2 / 5) 3 e -(2/5) 3! x 7.15e 3 T/5 T 25
(d) How much time should you allow so that with probability 0.99 at least oe bus arrives? at B P ( b least oe] 1) l0.01 ( T T B 0] 0.01 23mi ( T / 5) / 5) B 1] 1- B 0 e -(T/5) 1! e -(T/5) 0] 0.01 0.99 26
A Zipf (,α=1)r.v X has PMF P X c ( x) 0 / x x 1,2,..., else The costat C() is set so that Calculate c() for =1,2,,6 x1 P x X ( ) 1 for for for 1,, 2,, 3,, 1 x1 2 x1 3 x1 C(1) /1 1 C(2) / C(3) / x 1 x 1 C(1) 1 C(2) 1 C(3) 1 C(2) 2 C(3) 2 1 C(3) 3 C(2) 1 2 3 C(2) 27 6 11
A radio statio gives a pair of cocert tickets to the sixth caller who kows the birthday of the performer. For each perso who calls, the probability is 0.75 of kowig the birthday. All calls are idepedet. (a) What is the PMF of L,the umber of calls ecessary to fid the wier? Pascal R.V,, P L ( l) L 1 (0.75) 6 1 6 (1 (b) What is the probability of fidig the wier o the teth call? 0.75) L6 P L ( L 10) 101 (0.75) 6 1 6 (1 0.75) 106 0.0876 28
(b) What is the probability that the statio will eed ie or more calls to fid the wier? P L ( ie or more) P 1{ P L ( L 6) P L L ( L ( L 9) 7) P 1P L ( L L ( L 8)} 9) 29
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