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Chapter 4 Continuous Random Variables

Chapter 4 Continuous Random Variables
§ 4.1 Continuous Sample Space
n
The range of a discrete random variable is a countable set of numbers.
P[X=x] >= 0
(Probability Mass Function)
The range of a continuous random variable is an uncountable set of numbers.
There are infinite numbers between two limits
n
P[X=x] = 0
(Probability Mass Function)
When X is continuous, it is impossible to define a probability mass function PX(x).
1
§ 4.2 The Cumulative Distribution Function
The CDF FX(x) is a probability model for any random variable. The CDF FX(x) is a
continuous function if and only if X is a continuous random variable.
n
Definition 4.1 Cumulative Distribution Function (CDF)
FX(x)=P[X <=x]
Theorem 4.1
ForanyrandomvariableX.
𝑭𝑿 βˆ’βˆž = 𝟎
𝑭𝑿 ∞ = 𝟏
𝑷 π’™πŸ < 𝑿 < π’™πŸ = 𝑭𝑿 π’™πŸ βˆ’ 𝑭𝑿 π’™πŸ
ForanyrandomvariableX
X isacontinuousrandomvariableiftheCDFFX(x) isacontinuousfunction.
2
§ 4.3 Probability Density Function
n
The slope at any point x indicates the probability that X is near x.
P[x1 < X ≀ x1 + Ξ”] =
FX (x1 + Ξ”) βˆ’ FX (x1 )
Ξ”
Ξ”
Definition4.3ProbabilityDensityFunction(PDF)
f X (x) =
dFX (x)
dx
3
§ 4.3 Probability Density Function
Example4.5
⎧
βŽͺ
βŽͺβŽͺ
TheCDFofYis FY (y) = ⎨
βŽͺ
βŽͺ
βŽͺ⎩
0
y<0
y3 0 ≀ y ≀ 1
FindthePDFofYandprobabilitythatYis
between1/4and3/4
1 y >1
⎧
dFY (y) βŽͺ
f
(y)
=
=⎨
Solution: Y
dy
βŽͺ
⎩
3y 2
0 < y ≀1
0 otherwise
P[1 / 4 < Y ≀ 3 / 4] = FY (3 / 4) βˆ’ FY (1 / 4) = (3 / 4)3 βˆ’ (1 / 4)3 = 13 / 32
3/4
P[1 / 4 < Y ≀ 3 / 4] =
∫
1/4
3/4
fY (y)dy =
∫ 3y2 dy = 13 / 32
1/4
4
§ 4.3 Probability Density Function
Theorem4.2
ForacontinuousrandomvariableX withPDF f X (x)
(a) f X (x) β‰₯ 0 forallx
t
(b) FX (x) =
∫
f X (x)
βˆ’βˆž
(c)
∞
∫
f X (x) = 1
βˆ’βˆž
Theorem4.3
x2
P[x1 < X ≀ x2 ] =
∫ fX (x)dx
x1
5
§ 4.4 Expected Values
n
The expected value of a continuous random variable X is
0
𝑬 𝑿 = - 𝒙𝒇𝒙 𝒙 𝒅𝒙
10
Example 4.6
FindtheexpectedvalueofX,thePDFofXisgivenas
𝒇𝑿 𝒙 = 2
0
𝟏, 𝟎 ≀ 𝒙 < 𝟏
𝟎, π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
𝟏
𝑬 𝑿 = - 𝒙𝒇𝒙 𝒙 𝒅𝒙 = - 𝒙𝒅𝒙 = 𝟏/𝟐
10
𝟎
6
§ 4.4 Expected Values
Theorem 4.4
The expected value of a function, g(X), of a random variable X is
0
𝑬 π’ˆ(𝑿) = - π’ˆ(𝒙)𝒇𝒙 𝒙 𝒅𝒙
10
Example 4.8
1, 0 ≀ π‘₯ < 1
𝑓? π‘₯ = 2
LetXbeauniformrandomvariablewithPDFLetW=g(X)=0,ifX<=1/2,and
W=g(X)=1,
0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’
IfX>½.Findtheexpectedvalueofg(X).
0
Q
𝐸 𝑔(𝑋) = - 𝑔(π‘₯)𝑓? π‘₯ 𝑑π‘₯ = - π‘₯𝑑π‘₯
10
Q/R
7
§ 4.4 Expected Values
Theorem 4.5
For any random variable X:
𝒂 𝑬 𝑿 βˆ’ 𝝁𝑿 = 𝟎
Linearpropertiesofexpectedvalue
𝒃 𝑬 𝒂𝑿 + 𝒃 = 𝒂𝑬 𝑿 + 𝒃
𝒄 𝑽𝒂𝒓 𝑿 = 𝑬 π‘ΏπŸ βˆ’ ππŸπ‘Ώ
0
E 𝑋 R = ∫10 π‘₯ R 𝑓? π‘₯ 𝑑π‘₯
0
𝒄 𝑽𝒂𝒓 𝒂𝑿 + 𝒃 = π’‚πŸ 𝑬 π‘ΏπŸ
Var 𝑋 = ∫10 (π‘₯ βˆ’ πœ‡? )R 𝑓? π‘₯ 𝑑π‘₯
8
§ 4.5 Families of Continuous Random Variables
Definition 4.5 Uniform Random Variable
X is a uniform (a,b) random variable if the PDF of X is
𝒇𝑿 𝒙 = 2
𝟏/(𝒃 βˆ’ 𝒂), 𝒂 ≀ 𝒙 < 𝒃
𝟎,
π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
Theorem 4.6
If X is a uniform (a,b) random variable
𝒂 𝑻𝒉𝒆π‘ͺπ‘«π‘­π’π’‡π‘Ώπ’Šπ’”:
𝑭𝑿 𝒙 =
𝟎,
π’™βˆ’π’‚
,
π’ƒβˆ’π’‚
𝟏,
𝒙≀𝒂
𝒂<𝒙≀𝒃
𝒙>𝒃
𝒃 π‘»π’‰π’†π’†π’™π’‘π’†π’„π’•π’†π’…π’—π’‚π’π’–π’†π’π’‡π‘Ώπ’Šπ’”: 𝑬 𝑿 = (𝒃 + 𝒂)/𝟐
𝒃 π‘»π’‰π’†π’—π’‚π’“π’Šπ’‚π’π’„π’†π’π’‡π‘Ώπ’Šπ’”: Var 𝑿 = (𝒃 βˆ’ 𝒂)𝟐 /𝟏𝟐
9
§ 4.5 Families of Continuous Random Variables
Example 4.11
The phase angle, Θ, of the signal at the input to a modem is uniformly distributed between 0 and.
2Ο€
Radians. What are the PDF, CDF, expected value, and variance of Θ
π‘‡β„Žπ‘’π‘ƒπ·πΉπ‘œπ‘“π‘‹π‘–π‘ : 𝑓{ πœƒ = 2
1/2πœ‹, 0 ≀ π‘₯ < 2πœ‹
0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’
0,
π‘‡β„Žπ‘’πΆπ·πΉπ‘œπ‘“π‘‹π‘–π‘ : 𝐹} πœƒ =
π‘‡β„Žπ‘’π‘’π‘₯π‘π‘’π‘π‘‘π‘’π‘‘π‘£π‘Žπ‘™π‘’π‘’π‘œπ‘“π‘‹π‘–π‘ :
π‘‡β„Žπ‘’π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’π‘œπ‘“π‘‹π‘–π‘ :
π‘₯≀0
πœƒ
, 0 < π‘₯ ≀ 2πœ‹
2πœ‹
1, π‘₯ > 2πœ‹
𝐸 𝑋 =
Var 𝑋 =
2πœ‹ + 0
=πœ‹
2
t1u
QR
v
= πœ‹ w /3
10
§ 4.5 Families of Continuous Random Variables
Definition 4.6 Exponential Random Variable
X is an exponential (𝝀) random variable if the PDF of X is
𝒇𝑿 𝒙 = 2
𝝀𝒆1𝝀𝒙 , 𝒙 β‰₯ 𝟎
𝟎, π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
Theorem 4.8
If X is a exponential (𝝀) random variable
1𝝀𝒙
𝒂 𝑻𝒉𝒆π‘ͺπ‘«π‘­π’π’‡π‘Ώπ’Šπ’”: 𝑭𝑿 𝒙 = 2𝟏 βˆ’ 𝒆 , 𝒙 β‰₯ 𝟎
𝟎, π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
𝒃 π‘»π’‰π’†π’†π’™π’‘π’†π’„π’•π’†π’…π’—π’‚π’π’–π’†π’π’‡π‘Ώπ’Šπ’”: 𝑬 𝑿 = 𝟏/𝝀
𝒃 π‘»π’‰π’†π’—π’‚π’“π’Šπ’‚π’π’„π’†π’π’‡π‘Ώπ’Šπ’”: Var 𝑿 = 𝟏/π€πŸ
11
§ 4.5 Families of Continuous Random Variables
Example 4.12
The probability that a telephone call lasts no more than t minutes is often modeled as an
exponential CDF.
1 βˆ’ 𝑒 1β€š/w , 𝑑 β‰₯ 0
𝐹‒ 𝑑 = 2
0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’
What is the PDF of the duration in minutes of a telephone conversation? What is the probability
that a conversation will last between 2 and 4 minutes?
𝑑𝐹‒ (𝑑)
(1/3)𝑒 1β€š/w , 𝑑 β‰₯ 0
𝑓‒ 𝑑 =
=2
𝑑𝑑
0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’
𝑃 2 ≀ 𝑇 ≀ 4 = 𝐹‒ 4 βˆ’ 𝐹‒ 2 = (1 βˆ’ 𝑒
1
β€ž
w)
βˆ’ (1 βˆ’ 𝑒
1
β€ž
w)
= 0.250
12
§ 4.5 Families of Continuous Random Variables
Definition 4.7 Erlang Random Variable
X is an Erlang (n, 𝝀) random variable if the PDF of X is
𝒇𝑿
𝝀𝒏 𝒙𝒏1𝟏 𝒆1𝝀𝒙
𝒙β‰₯𝟎
𝒙 =‑ π’βˆ’πŸ ! ,
π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
𝟎,
Wheretheparameterπ›Œ > 𝟎, π’‚π’π’…π’Šπ’π’•π’†π’ˆπ’†π’“π’ β‰₯ 𝟏
Theorem 4.8
If X is a Erlang (n, 𝝀) random variable
𝒏1𝟏 (𝝀𝒙)π’Œ 𝒆1𝝀𝒙
, 𝒙β‰₯𝟎
𝒂 𝑻𝒉𝒆π‘ͺπ‘«π‘­π’π’‡π‘Ώπ’Šπ’”: 𝑭𝑿 𝒙 = β€‘πŸ βˆ’ ‰ π’Œβ€ΉπŸŽ
π’Œ!
𝟎, π’π’•π’‰π’†π’“π’˜π’Šπ’”π’†
𝒃 π‘»π’‰π’†π’†π’™π’‘π’†π’„π’•π’†π’…π’—π’‚π’π’–π’†π’π’‡π‘Ώπ’Šπ’”: 𝑬 𝑿 = 𝒏/𝝀
𝒃 π‘»π’‰π’†π’—π’‚π’“π’Šπ’‚π’π’„π’†π’π’‡π‘Ώπ’Šπ’”: Var 𝑿 = 𝒏/π€πŸ
13
§ 4.6 Gaussian Random Variables
Definition 4.8 Gaussian Random Variable
X is an Gaussian (𝝁, 𝝈) random variable if the PDF of X is
𝒇𝑿 𝒙 =
𝟏
πŸπ…πˆπŸ
𝒆1
𝒙1𝝁
𝟐 /𝟐𝝈 𝟐
Wheretheparameter𝝁 canbeanyrealnumber,andtheparameter𝝈 > 𝟎,
14
§ 4.6 Gaussian Random Variables
Theorem 4.12
If X is a Gaussian (𝝁, 𝝈) random variable
𝒃 π‘»π’‰π’†π’†π’™π’‘π’†π’„π’•π’†π’…π’—π’‚π’π’–π’†π’π’‡π‘Ώπ’Šπ’”: 𝑬 𝑿 = 𝝁
𝒃 π‘»π’‰π’†π’—π’‚π’“π’Šπ’‚π’π’„π’†π’π’‡π‘Ώπ’Šπ’”: Var 𝑿 = 𝝈𝟐
Theorem 4.13
If X is a Gaussian (𝝁, 𝝈) random variable, Y = aX + b is a Gaussian (𝐚𝝁+b, a𝝈)
15
§ 4.6 Gaussian Random Variables
Definition 4.9 Standard Normal Random Variable
The standard normal random variable Z is the Gaussian (𝟎, 𝟏) random variable
π‘»π’‰π’†π’†π’™π’‘π’†π’„π’•π’†π’…π’—π’‚π’π’–π’†π’π’‡π’π’Šπ’”: 𝑬 𝒁 = 𝟎
π‘»π’‰π’†π’—π’‚π’“π’Šπ’‚π’π’„π’†π’π’‡π’π’Šπ’”: Var 𝒁 = 𝟏
Definition 4.10 Standard Normal CDF
The CDF of the standard normal random variable Z is
π›Ÿ 𝒛 =
𝟏
πŸπ…
𝒛
- 𝒆1𝒖
𝟐 /𝟐
𝒅𝒖
10
16
§ 4.6 Gaussian Random Variables
Theorem 4.14
If X is a Gaussian (𝝁, 𝝈) random variable, the CDF of X is
𝑭𝑿 𝒙 = π›Ÿ
π’™βˆ’π
𝝈
The probability that X is in the interval (a, b] is:
𝑷𝒂<𝑿≀𝒃 =π›Ÿ
π’ƒβˆ’π
π’‚βˆ’π
βˆ’π›Ÿ
𝝈
𝝈
Usingthistheorem,wetransformvaluesofaGaussianrandomvariable,X,toequivalentvaluesofthestandard
normalrandomvariable,Z.Forasamplevaluex oftherandomvariableX,thecorrespondingsamplevalueofZ is
𝒛=
π’™βˆ’π
𝝈
17
§ 4.6 Gaussian Random Variables
Example 4.15
Suppose your score on a test is x = 46, a sample value of the Gaussian (61, 10) random
variable. Express your test score as a sample value of the standard normal random variable, Z.
𝒛=
𝒛=
π’™βˆ’π
𝝈
πŸ’πŸ”1πŸ”πŸ
𝟏𝟎
=-1.5
18
§ 4.6 Gaussian Random Variables
Theorem 4.15
π›Ÿ βˆ’π’› = 𝟏 βˆ’ π›Ÿ 𝒛
Example 4.16
If X is a Gaussian (61, 10) random variable, what is P[X<=46]?
ApproachI:
BasedonTheorem4.14:𝑃 𝑋 ≀ 𝑏 = Ο•
t1β„’
Ε‘
=Ο•
β€žβ€Ί1β€ΊQ
QΕ“
= Ο• βˆ’1.5 = 1 βˆ’ Ο• 1.5 = 1 βˆ’ 0.933 = 0.067
ApproachII:
BasedonTheorem4.1:𝑃 π‘₯Q < 𝑋 < π‘₯R = 𝐹? π‘₯R βˆ’ 𝐹? π‘₯Q = 𝐹? 46 =
β€žβ€Ί
Q
∫10
RΕΈQΕ“ v
𝑒
1
¡¢£ v
vβˆ—£¥ v
𝑑π‘₯ = 0.067
19
§ 4.6 Gaussian Random Variables
Definition 4.11 Standard Normal Complementary CDF
The standard normal complementary CDF is
𝑸 𝒛 =𝑷𝒁>𝒛 =
𝟏
πŸπ…
0
- 𝒆1𝒖
𝟐 /𝟐
𝒅𝒖 = 𝟏 βˆ’ 𝚽(𝒛)
𝒛
20
§ 4.6 Gaussian Random Variables
Quiz 4.6
X is the Gaussian (0,1) random variable and Y is the Gaussian (0, 2) random variable. Sketch
the PDFs fX(x) and fY(y) on the same axes and find:
(a)P[-1<X<=1]
(b)P[-1<Y<=1]
(c)P[X>3.5]
(d)P[Y>3.5]
21
§ 4.7 Delta Functions, Mixed Random Variables
Definition 4.12 Unit impulse (Delta) function
𝑑¨ π‘₯ = 2
1/πœ€
βˆ’πœ€/2 ≀ π‘₯ ≀ πœ€/2
0 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’
Theunitimpulsefunctionis 𝛿 π‘₯ = lim 𝑑¨ (π‘₯)
¨β†’Ε“
Theorem 4.16 (sifting property)
0
- 𝑔 π‘₯ 𝛿 π‘₯ βˆ’ π‘₯Ε“ 𝑑π‘₯ = 𝑔(π‘₯Ε“ )
10
22
§ 4.7 Delta Functions, Mixed Random Variables
Definition 4.13 Unit step function
u π‘₯ =2
0
1
π‘₯<0
π‘₯β‰₯0
Theorem 4.17 (construct unit step function from delta function)
¯
- 𝛿 π‘₯ 𝑑π‘₯ = 𝑒(π‘₯)
10
𝛿 π‘₯ =
𝑑𝑒(π‘₯)
𝑑π‘₯
23

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