Important Discrete Distributions
Transcript of Important Discrete Distributions
Iyer - Lecture 10 ECE 313 โ Fall 2016
Important Discrete Distributions: Poisson, Geometric, & Modified Geometric
ECE 313Probability with Engineering Applications
Lecture 10Ravi K. Iyer
Department of Electrical and Computer Engineering University of Illinois
Iyer - Lecture 10 ECE 313 โ Fall 2016
Todayโs Topics
โขRandom VariablesโExample: โGeometric/modified Geometric Distribution โ Poisson derived from Bernoulli trials; ExamplesโExamples: Verifying CDF/pdf/pmf;
โขAnnouncements:โ Homework 4 out todayโ In class activity next Wednesday,.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Binomial Random Variable (RV) Example:Twin Engine vs 4-Engine Airplane
โข Suppose that an airplane engine will fail, when in flight, withprobability 1โp independently from engine to engine. Also,suppose that the airplane makes a successful flight if at least 50percent of its engines remain operative. For what values of p isa four-engine plane preferable to a two-engine plane?
โข Because each engine is assumed to fail or functionindependently: the number of engines remaining operational isa binomial random variable. Hence, the probability that a four-engine plane makes a successful flight is:
4322
04322
)1(4)1(6
)1(44
)1(34
)1(24
ppppp
pppppp
+-+-=
-รทรทรธ
รถรงรงรจ
รฆ+-รทรท
รธ
รถรงรงรจ
รฆ+-รทรท
รธ
รถรงรงรจ
รฆ
Iyer - Lecture 10 ECE 313 โ Fall 2016
Binomial RV Example 3 (Contโ)โข The corresponding probability for a two-engine plane is:
โข The four-engine plane is safer if:
โข Or equivalently if:
โข Hence, the four-engine plane is safer when the engine successprobability is at least as large as 2/3, whereas the two-engine plane issafer when this probability falls below 2/3.
24322 )1(2)1(4)1(6 pppppppp +-ยณ+-+-
22 )1(222
)1(12
pppppp +-=รทรทรธ
รถรงรงรจ
รฆ+-รทรท
รธ
รถรงรงรจ
รฆ
pppppp -ยณ+-+- 2)1(4)1(6 322
0)23()1(02783 223 ยณ--ยณ-+- pporppp
32023 ยณยณ- porp
Iyer - Lecture 10 ECE 313 โ Fall 2016
Geometric Distribution: Examples
โข Some Examples where the geometric distribution occurs
1. The probability the ith item on a production line is defective is given by the geometric pmf.
2. The pmf of the random variable denoting the number of time slices needed to complete the execution of a job
Iyer - Lecture 10 ECE 313 โ Fall 2016
Geometric Distribution Examples
3. Consider a repeat loop
โข repeat S until B
โข The number of tries until B (success) is reached (i.e.,includesB), is a geometrically distributed Random Variable with parameter p.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Discrete DistributionsGeometric pmf (cont.)
โข To find the pmf of a geometric Random Variable (RV), Z note that the event [Z = i] occurs if and only if we have a sequence of (i โ 1) โfailuresโ followed by one success - a sequence of independent Bernoulli trials each with the probability of success equal to p and failure q.
โข Hence, we have the pdffor i = 1, 2,..., (A)
โ where q = 1 - p.โข Using the formula for the sum of a geometric series, we have:
โข CDF of Geometric distr.:
1i1iZ p)p(1pq(i)p -- -==
11
)(1 1
1 ==-
==รฅ รฅยฅ
=
ยฅ
=
-
pp
qppqip
i i
iZ
รซ รปรซ รป
รฅ=
- --=-=t
t
1i
1iZ p)(11p)p(1(t)F
Iyer - Lecture 10 ECE 313 โ Fall 2016
Modified Geometric Distribution Example
โข Consider the program segment consisting of a while loop:
โข while ยฌ B do S
โข the number of times the body (or the statement-group S) of the loop is executed: a modified geometric distribution with parameter p (probability the B is not true) โ no. of failures until the first success.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Discrete Distributionsthe Modified Geometric pmf (cont.)
โข The random variable X is said to have a modified geometric pmf, specify by
for i = 0, 1, 2,...,
โข The corresponding Cumulative Distribution function is:
iX p)p(1(i)p -=
for t โฅ 0รซ รป
รซ รป
รฅ=
+--=-=t
i
tiX ppptF
0
1)1(1)1()(
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: Geometric Random Variable
A representative from the NFL Marketing division randomly selects people on a random street in Chicago loop, until he/she finds a person who attended the last home football game. Let p, the probability that she succeeds in finding such a person, is 0.2 and X denote the number of people asked until the first success. โข What is the probability that the representative must select 4
people until he finds one who attended the last home game?๐ ๐ = 4 = 1 โ 0.2 * 0.2 = 0.1024
โข What is the probability that the representative must select more than 6 people before finding one who attended the last home game?๐ ๐ > 6 = 1 โ ๐ ๐ โค 6 = 1 โ 1 โ 1 โ 0.2 . = 0.262
Iyer - Lecture 10 ECE 313 โ Fall 2016
The Poisson Random Variableโข A random variable X, taking on one of the values 0,1,2,โฆ, is said
to be a Poisson random variable with parameter ฮฑ, if for some ฮฑ>0,
defines a probability mass function since
1!
)(00
=== -ยฅ
=
ยฅ
=
- รฅรฅ aaa a eei
eipi
i
i
,...1,0,!
}{)( ==== - ii
eiXPipiaa
Iyer - Lecture 10 ECE 313 โ Fall 2016
Poisson Random Variable
Consider smaller intervals, i.e., let ๐ โ โ
๐ ๐ = ๐ = lim6โ7
๐๐ก:
๐!๐ ๐ โ 1 โฆ (๐ โ ๐ + 1)
๐ @ ๐ @ ๐โฆ๐ 1 โ๐๐ก๐
61 โ
๐๐ก๐
A:
= lim6โ7
๐๐ก:
๐! 1 @ 1 โ1๐ 1 โ
2๐ โฆ (1 โ
๐ + 1๐ ) 1 โ
๐๐ก๐
61 โ
๐๐ก๐
A:
= BCD
:!๐ABC
Which is a Poisson process with ๐ผ = ๐๐ก
Iyer - Lecture 10 ECE 313 โ Fall 2016
Geometric Distribution Examples
3. Consider the program segment consisting of a while loop:
โข while ยฌ B do S
โข the number of times the body (or the statement-group S) of the loop is executed: a modified geometric distribution with parameter p (probability the B is not true) โ no. of failures until the first success.
4. Consider a repeat loop
โข repeat S until B
โข The number of tries until B (success) is reached will be a geometrically distributed random variable with parameter p.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Discrete DistributionsGeometric pmf (cont.)
โข To find the pmf of Z note that the event [Z = i] occurs if and only if we have a sequence of (i โ 1) failures followed by one success - a sequence of independent Bernoulli trials each with the probability of success equal to p and failure q.
โข Hence, we havefor i = 1, 2,..., (A)
โ where q = 1 - p.โข Using the formula for the sum of a geometric series, we have:
โข CDF of Geometric distr.:
1i1iZ p)p(1pq(i)p -- -==
11
)(1 1
1 ==-
==รฅ รฅยฅ
=
ยฅ
=
-
pp
qppqip
i i
iZ
รซ รปรซ รป
รฅ=
- --=-=t
t
1i
1iZ p)(11p)p(1(t)F
Iyer - Lecture 10 ECE 313 โ Fall 2016
Discrete Distributionsthe Modified Geometric pmf (cont.)
โข The random variable X is said to have a modified geometric pmf, specify by
for i = 0, 1, 2,...,
โข The corresponding Cumulative Distribution function is:
iX p)p(1(i)p -=
for t โฅ 0รซ รป
รซ รป
รฅ=
+--=-=t
i
tiX ppptF
0
1)1(1)1()(
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: Geometric Random Variable
A representative from the NFL Marketing division randomly selects people on a random street in Chicago loop, until he/she finds a person who attended the last home football game. Let p, the probability that he succeeds in finding such a person, be 0.2 and X denote the number of people he selects until he finds his first success. โข What is the probability that the representative must select 4
people until he finds one who attended the last home game?๐ ๐ = 4 = 1 โ 0.2 * 0.2 = 0.1024
โข What is the probability that the representative must select more than 6 people before he finds one who attended the last home game?๐ ๐ > 6 = 1 โ ๐ ๐ โค 6 = 1 โ 1 โ 1 โ 0.2 . = 0.262
Iyer - Lecture 10 ECE 313 โ Fall 2016
The Poisson Random Variableโข A random variable X, taking on one of the values 0,1,2,โฆ, is said
to be a Poisson random variable with parameter ฮป, if for some ฮป>0,
defines a probability mass function since
1!
)(00
=== -ยฅ
=
ยฅ
=
- รฅรฅ lll l eei
eipi
i
i
,...1,0,!
}{)( ==== - ii
eiXPipill
Iyer - Lecture 10 ECE 313 โ Fall 2016
Poisson Random Variable
Consider smaller intervals, i.e., let ๐ โ โ
๐ ๐ = ๐ = lim6โ7
๐๐ก:
๐!๐ ๐ โ 1 โฆ (๐ โ ๐ + 1)
๐ @ ๐ @ ๐โฆ๐ 1 โ๐๐ก๐
61 โ
๐๐ก๐
A:
= lim6โ7
๐๐ก:
๐! 1 @ 1 โ1๐ 1 โ
2๐ โฆ (1 โ
๐ + 1๐ ) 1 โ
๐๐ก๐
61 โ
๐๐ก๐
A:
= BCD
:!๐ABC
Which is a Poisson process with ๐ผ = ๐๐ก
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: Poisson Random Variables
A cloud computing system failure occurs according to a Poisson distribution with an average of 3 failures every 10 weeks. I.e., the failure within t week(s) is Poisson distributed with ๐ผ= 0.3ti. Calculate the probability that there will not be more than one
failure during a particular week
๐ฟ๐๐ก๐C๐๐๐๐๐ก๐๐กโ๐๐๐ข๐๐๐๐๐๐๐๐๐๐๐ข๐๐๐ ๐ค๐๐กโ๐๐๐ก๐ค๐๐๐๐
๐ ๐V = 0๐๐1 = ๐ ๐V = 0 + ๐ ๐V = 1 =๐AW.*0.3W
0! +๐AW.*0.3V
1!ii. Calculate the probability that there will be at least one failure
during a particular month (4 weeks)
๐ ๐Y โฅ 1 = 1 โ ๐ ๐Y = 0 = 1 โ๐AW.*โY(0.3 โ 4)W
0!
Iyer - Lecture 10 ECE 313 โ Fall 2016
Poisson Random Variable
A manufacturer produces VLSI chips, 1% of which are defective. Find the probability that in a box containing 100 chips, no defectives are found.
Using Poisson approximation, ฮฑ=100*0.01=1,
We can verify that the probabilities are non negative and sum to 1
]ฮฑ:
๐! ๐Aฮฑ
7
:^W
= ๐Aฮฑ]ฮฑ:
๐!
7
:^W
= ๐Aฮฑ๐ฮฑ = 1
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: PDF, PMF
Verify whether below are valid PDF/PMF.
๐ ๐ฅ = `VY๐ฅ*, 0 โค ๐ฅ โค 20, ๐๐กโ๐๐๐ค๐๐ ๐
Answer:1. ๐ ๐ฅ โฅ 02. โซ ๐ ๐ฅ ๐๐ฅ = โซ V
Y๐ฅ*๐๐ฅ = 1c
W7A7
Therefore, it is a valid PDF.
๐ ๐ฅ = eVVW
3๐ฅ โ 2 , ๐ฅ = 0,1,2,30, ๐๐กโ๐๐๐ค๐๐ ๐
Answer:1. โ ๐7
:^W (x=k)=โ ๐*:^W (x=k)=1
2. However, p(x=0)=AcVW< 0
Therefore, it is not a valid PMF.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: CDF
โข Verify whether the following is a valid CDF
๐น ๐ฅ = e0, ๐ฅ < 0
๐ฅc, 0 โค ๐ฅ < 11, ๐ฅ โฅ 1
Answer:1. lim
rโA7๐น ๐ฅ = 0
2. limrโ7
๐น ๐ฅ = 13. IsF(x)monotonicallynon-decreasing? Yes
Therefore, it is a valid CDF.
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: Identify Random Variables
For each of the following random variables, i) determine if it is discrete or continuous, ii) specify the distribution of the identified random variable (from those you have learned in the class), iii) write the formula for probability mass function (pmf) or probability density function (pdf) based on the information provided:
I. A binary communication channel introduces a single bit error in each transmission with probability of 0.1. Let X be the random variable representing the number of errors in n independent transmissions.
i. Discreteii. Binomialiii. ๐ ๐ = ๐ = 6
๏ฟฝ ๐๏ฟฝ 1 โ ๐ 6A๏ฟฝ
Iyer - Lecture 10 ECE 313 โ Fall 2016
Example: Identify Random Variables
II. A sequence of characters is transmitted over a channel that introduces errors with probability 0.2. Let N be the random variable representing the number of error-free characters transmitted before an error occurs.
i. Discreteii. Modified Geometriciii. ๐ ๐ = ๐ = 1 โ ๐ ๏ฟฝ๐
Iyer - Lecture 10 ECE 313 โ Fall 2016
Review: Continuous Random Variables
โข Continuous Random Variables:โ Probability distribution function (pdf):
โข Properties:
โข All probability statements about X can be answered by f(x):
โ Cumulative distribution function (CDF):
โข Properties:โข A continuous function
P{X โ B} = f (x)dxBโซ
รฒยฅ
ยฅ-=ยฅ-ยฅร= dxxfXP )()},({1
รฒ=ยฃยฃb
adxxfbXaP )(}{
รฒ ===a
adxxfaXP 0)(}{
)()( afaFdad
=
รฒยฅ-
ยฅ<<ยฅ-=ยฃ=x
xx xdttfxXPxF , )()()(
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution
โข Extremely important in statistical application because of the central limit theorem:โ Under very general assumptions, the mean of a sample of n mutually
independent random variables is normally distributed in the limit n ยฎยฅ.โข Errors in measurement often follows this distribution.โข During the wear-out phase, component lifetime follows a normal
distribution.โข The normal density is given by:
where are two parameters of the distribution.
f (x) = 1ฯ 2ฯ
exp โ12x โยตฯ
"
#$
%
&'
2(
)**
+
,--, โโ < x <โ
โโ < ยต <โ and ฯ > 0
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution (cont.)
โข Normal density with parameters ยต =2 and s =1
f x(x)
0.4
0.2
-2.00 1.00 4.00 7.00-5.00
x
s =1
ยต = 2
1ฯ 2ฯ
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution (cont.)
โข The distribution function (CDF) F(x) has no closed form, so between every pair of limits a and b, probabilities relating to normal distributions are usually obtained numerically and recorded in special tables.
โข These tables apply to the standard normal distribution[Z ~ N(0,1)] --- a normal distribution with parameters ยต = 0 , s = 1 --- and their entries are the values of:
รฒยฅ-
-=z
tZ dtezF 2
2
21)(p
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution (cont.)
โข Since the standard normal density is clearly symmetric, it follows that for z > 0:
โข The tabulations of the normal distribution are made only for z โฅ 0 To find P(a โค Z โค b), use F(b) - F(a).
FZ (โz) = fZ (t)dtโโ
โ z
โซ
= fZz
โ
โซ (โt)dt
= fZz
โ
โซ (t)dt
= fZ (t)dt โ fZโโ
z
โซโโ
โ
โซ (t)dt
= 1โ FZ (z)
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution (cont.)
โข The CDF of the N(0,1) distribution ( ) is denoted in the tables by , and its complementary CDF is denoted by , so: F Q
)(zFZ
)()(1)( uuuQ -F=F-=
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution (cont.)
โข For a particular value, x, of a normal random variable X, the corresponding value of the standardized variable Z is:
โข The Cumulative distribution function of X can be found by using:
alternatively:
โข Similarly, if X is normally distributed with parameters ฮผ and ฯ2 thenZ = ฮฑX + ฮฒ is normally distributed with parameters ฮฑฮผ + ฮฒ and ฮฑ2ฯ2.
sยต /)( -= XZ
)( )(
)(
)()(
sยตsยต
sยต
zFzXP
zXP
zZPzF
X
Z
+=+ยฃ=
ยฃ-
=
ยฃ=
)()(sยต-
=xFxF ZX
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution Example 1
โข An analog signal received at a detector (measured in microvolts) may be modeled as a Gaussian random variable N(200, 256) at a fixed point in time. What is the probability that the signal will exceed 240 microvolts? What is the probability that the signal is larger than 240 microvolts, given that it is larger than 210 microvolts?
P(X > 240) = 1โ P(X โค 240)
= 1โ FZ240 โ 200
16# $ %
& ' (
= 1โ FZ (2.5) โ 0.00621
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution Example 1 (cont.)
โข Next:
P(X โฅ 240 | X โฅ 210) =P(X) โฅ 240)P(X โฅ 210)
=1โ FZ
240 โ 20016
# $ %
& ' (
1โ FZ210 โ 200
16# $ %
& ' (
=0.006210.26599
โ 0.02335
Iyer - Lecture 10 ECE 313 โ Fall 2016
Normal or Gaussian Distribution Example 2
โข Assuming that the life of a given subsystem, in the wear-out phase, is normally distributed with ยต = 10,000 hours and s = 1,000 hours, determine the reliability for an operating time of 500 hours given that
โ (a) The age of the component is 9,000 hours,โ (b) The age of the component is 11,000.
โข The required quantity under (a) is R9,000(500) and under (b) is R11,000(500).