continuous random variable theory.pdf
Post on 23-Feb-2018
245 Views
Preview:
TRANSCRIPT
-
7/24/2019 Continuous random variable theory.pdf
1/101
(Continuous Random Variable )
Continuous Densities:
-
7/24/2019 Continuous random variable theory.pdf
2/101
X can assume all possible values 0x1
Since possible values of X are non countablein 0x1, then what happens to point
probabilities?
-
7/24/2019 Continuous random variable theory.pdf
3/101
Definition:
A random variable is continuous if it can
assume any value in some interval
( or intervals ) of real numbers and the
probability that it assume any specific value
is 0 ( zero ).
-
7/24/2019 Continuous random variable theory.pdf
4/101
CONTINUOUS PDF (Density Function)
Definition:Let X be a continuous randomvariable. A function f(x) is called continuousdensity ( probability density function i.e pdf ) if
1 . f ( x ) 0
2 . ( ) 1f x d x
3. For anyab we havec
ad xxf )(c)xP(a.3
-
7/24/2019 Continuous random variable theory.pdf
5/101
(Thus P[aXc] is area under graph of y=f(x)
betweenx =a andx =c.)
-
7/24/2019 Continuous random variable theory.pdf
6/101
Remark: It is a consequence of the definition
that for any specified value of X, say xo , wehave P[X=xo] = 0, since
[ ] ( ) 0
o
o
x
o
xP X x f x d x
Remark :f (x 0)P [X=x 0].
In factf is analogous to the density of mass
whereas probability is analogous to the mass
itself.
-
7/24/2019 Continuous random variable theory.pdf
7/101
Remark. If X assumes values in some finiteinterval [a,c], we simply set f(x) = 0 for all
x[a,c].
-
7/24/2019 Continuous random variable theory.pdf
8/101
Every conceivable point on the line segment
could be the outcome of the experiment.Since X is continuous r.v, we have
][
][
][][
cxaP
cxaP
cxaPcxaP
-
7/24/2019 Continuous random variable theory.pdf
9/101
Let X be the continuous r.v. with density
f(x). The cumulative distribution function (cdf ) for X ,denoted by F(X) , is defined by
F(X) = P ( X x ) , all x
=
PROBABILITY by using cdf:F(x)
P( aXc ) = F(c) F(a).
x
ds)s(f
-
7/24/2019 Continuous random variable theory.pdf
10/101
Example : If pdf of a random variable X is
otherwise0
3x11)-(xr
1x0)1(
)(
xr
xf
(i) Find r and graph of f(x) (ii) Find F(x)
-
7/24/2019 Continuous random variable theory.pdf
11/101
3x1
3x15
)1(
5
1
1x0)2
(52
)(
2
2
x
xx
xF
-
7/24/2019 Continuous random variable theory.pdf
12/101
Theorem: Let F be the continuous cdf of a
continuous r.v with pdf f, then
abledifferenti
isFwhichatxallfor
),()( xFd x
d
xf
-
7/24/2019 Continuous random variable theory.pdf
13/101
(5) (Continuous uniform distribution) A
random variable X is said to be uniformlydistributed over an interval (a,c) if its
density is given by
cxaac
xf
,1)(
(a)Show that this is a density for a continuousrandom variable.
Sol: acf or
ac
Since
,01
-
7/24/2019 Continuous random variable theory.pdf
14/101
Secondly,
11
d xac
c
a
(b) Sketch the graph of the uniform density.
X=a X=c
f(x)
x
1/(c-a)
-
7/24/2019 Continuous random variable theory.pdf
15/101
( ii )Shade the area in the graph of part (b)
that represents P[Xa+c)/2].
X=a X=c
f(x)
x(a+c)/2
-
7/24/2019 Continuous random variable theory.pdf
16/101
(c ) Find the probability pictured in part: ii
(e) Let (l,m) and (d,f) be subintervals of (a,c)
of equal length. What is the relationship
between P[lX m] and P[dX f]
Sol: Probability is same on equal
length of interval.
]c
c
-
7/24/2019 Continuous random variable theory.pdf
17/101
(10) Find the general expression for the
cumulative uniform distribution for arandom variable X over (a,c)
cxaacax
d sac
xXPxFx
a
,
)(
1][)(
-
7/24/2019 Continuous random variable theory.pdf
18/101
cx
cxaac
ax
ax
xF
,1
,
,0
)(
-
7/24/2019 Continuous random variable theory.pdf
19/101
Section 4.2
Def: Let X be a continuous random variable
with pdf f. The expected value of X is defined
as
[ ] ( ) .E X x f x d x
Again E(x) exists if and only if
| | ( )x f x d x is f inite
-
7/24/2019 Continuous random variable theory.pdf
20/101
For a random variable X which is a function,
say H(x), the definition takes the form:
[ ( )] ( ) ( ) .E H x H x f x d x
provided
| ( ) | ( )H x f x d x is f inite
-
7/24/2019 Continuous random variable theory.pdf
21/101
Moment generating function :
( ) [ ]
( )
w here is densi ty o f .
tXX
tx
m t E e
e f x d x
f X
-
7/24/2019 Continuous random variable theory.pdf
22/101
Variance :
Variance is shape parameter in the
sense that a random variable with
small variance will have a compact
density; one with a large variance will
have a density that is rather spread
out or flat.
-
7/24/2019 Continuous random variable theory.pdf
23/101
MGF of uniform distribution random
variable on (a, c) :
ace
t
ac
d x
eeEta
c
a
txtx
tc
e1
)(
-
7/24/2019 Continuous random variable theory.pdf
24/101
Either by using mgf or directly, mean and
variance can be found.
12)()(
2)(
2
acXVar
caXE
-
7/24/2019 Continuous random variable theory.pdf
25/101
Section 4.3:
Definition : The Gamma function is the functiondefined by
1
0
1 1
1
( ) , 0.
1
lim lim0
exists for 0.
z
z z
r
z e d z
s
z e d z z e d zr s
-
7/24/2019 Continuous random variable theory.pdf
26/101
4.3: Gamma Random variable
Gamma Function: which is an improper
integral allows us to define exponential
and chi- squared random variables.
Gamma function:
0
1 0,)( d zez z
-
7/24/2019 Continuous random variable theory.pdf
27/101
Theorem: (Properties of Gamma function)
1),1()1()(.2
1)1(.1
allf orProof:by definition of Gamma function, we have
0 0
zz0
1dzedzez)1(
by integration by parts, we have
-
7/24/2019 Continuous random variable theory.pdf
28/101
)1()1(
)1(|
0,)(
0
1)1(
0
1
0
1
d zzeze
d zez
zz
z
Hint: by repeated use of L hospital rule,
we shall have
-
7/24/2019 Continuous random variable theory.pdf
29/101
)!1()1()2()1()2()2)(1(
)1()1()(
,
)!1()(
nnnnnn
nnn
Since
nn
Thus, Gamma function is a generalization of the
Factorial notation
-
7/24/2019 Continuous random variable theory.pdf
30/101
provedbenot to
)21(
0
2/1
d zez z
-
7/24/2019 Continuous random variable theory.pdf
31/101
GAMMA RANDOM VARIABLE
A random variable X with density function
is said to have a Gamma Distribution with
parameters and,for x>0,>0,>0.
wiseother
xexxf
x
,0
0,0,0,)(
1
)(
/1
-
7/24/2019 Continuous random variable theory.pdf
32/101
To check the necessary and sufficient condition
of pdf: 0xallfor0)x(f
0
11
0
/1
)(
1
,
)(
1)(,
d zez
zxandd zd xzx
Let
d xexd xxfFurther
z
x
-
7/24/2019 Continuous random variable theory.pdf
33/101
p d faisxfH ence
ez z
)(
1
)( 0
1
Theorem: Let X be a gamma random variable
with parameters and. Then m.g.f for X isgiven by:
2
x
)x(Var.3
]X[E.2
1t,)t1()t(m.1
-
7/24/2019 Continuous random variable theory.pdf
34/101
Proof:
)1(
)1(
)1(
)(1
)(
1
][)(,
0
)1
(1
/1
0
t
d zd xand
t
zx
xtzlet
d xex
d xexe
eEtmd efby
xt
xtx
tx
x
-
7/24/2019 Continuous random variable theory.pdf
35/101
1,)1()(
)()1()(
1
)1()(
1)1(
)
1
(
)(
1
0
1
1
0
tttm
t
d zezt
t
d ze
t
z
x
z
z
-
7/24/2019 Continuous random variable theory.pdf
36/101
0t
1
0tx
/)()t1(
/)t(m
dt
d]X[E.2
2
02
22
)1(
/))((][
tx tmd td
XE
-
7/24/2019 Continuous random variable theory.pdf
37/101
0
2 12
2
00
2
2 2 2
( )2 ) [ ]
( ) (1 )3 ) [ ]
( 1)
T h u s [ ] ( 1) ( ) .
X
t
X
tt
d m tE Xd t
d m t d tE X
d t d t
V a r X
wiseother
xex
xf
x
,0
0,0,0,
)(
1
)(
/1
-
7/24/2019 Continuous random variable theory.pdf
38/101
For >1 maximumValue of density is at
x=(-1)
Gamma(2 ,3) Gamma(2 0,0.5)
Gamma(1,4)
-
7/24/2019 Continuous random variable theory.pdf
39/101
For >1 maximum
Value of gamma densityis at
x=(-1)
2
)(.
][.
1,)1()(.
xVariii
XEii
tttmi x
-
7/24/2019 Continuous random variable theory.pdf
40/101
Exponential distribution : exponential
random variable is Gamma random
variable with=1.The density is
> 0 is the parameter of this exponentialdistribution. E[X]=, Var[X]=2 .
otherwise0
00, x1)(
x
exf
-
7/24/2019 Continuous random variable theory.pdf
41/101
The c.d.f. of exponential distribution with
Parameter is given by
o x1
111)(
0
0
0
xxs
x
s
x s
ee
ed sexF
-
7/24/2019 Continuous random variable theory.pdf
42/101
/1 , 0
( ) 0, otherwise.
xe x
F x
-
7/24/2019 Continuous random variable theory.pdf
43/101
Moment generating function, Mean and
Variance of exponential distribution
Note: Put =1 in the gamma distribution
we get the required results.
2
1
)(][
)1()(
XVarmeanXE
ttmX
-
7/24/2019 Continuous random variable theory.pdf
44/101
Poisson Process and Exponential dist :
For a Poisson process with parameter ,
the
w iting time
W is the time in the
given interval before the1
st
success.
Theorem :W has an exponential
distribution with parameter =1/.
-
7/24/2019 Continuous random variable theory.pdf
45/101
Proof: This theorem is distribution of
waiting time
The distribution function F for W is given by
]wW[P1]wW[P)w(F
Here, we have that the first occurrence of the
event will take place after time w only if number
of occurrences in the time interval [0,w] is zero
Let X be the number of occurrences of
the event in this time interval [0,w].
-
7/24/2019 Continuous random variable theory.pdf
46/101
X is a poisson random variable with parameter
w.
w
0w
e!0
)w(e
]0X[P]wW[P,Thus
w
ewWPwF
1][1)(
-
7/24/2019 Continuous random variable theory.pdf
47/101
Since, in the continuous case, the derivative
of the cumulative distribution function isthe density
1with
varibalerandomlexponentia
anfordensityexactlyisThis
)()(
wewfwF
-
7/24/2019 Continuous random variable theory.pdf
48/101
GAMMA RANDOM VARIABLE
A random variable X with density function
is said to have a Gamma Distribution with
parameters and,for x>0,>0,>0.
wiseother
xexxf
x
,0
0,0,0,)(
1
)(
/1
-
7/24/2019 Continuous random variable theory.pdf
49/101
Chi-square distribution : If a random variable
X has a gamma distribution with parameters=2 and=/2 , then X is said to have achi-square ( 2 ) distribution with degrees ofFreedom and denoted by X2
, is a positive
Integer.
-
7/24/2019 Continuous random variable theory.pdf
50/101
0x,ex2)
2
(1)x(f 2/x
1
2
2/
E[X2 ]=, Var[X2 ]=2
=2 and=/2 ,
-
7/24/2019 Continuous random variable theory.pdf
51/101
-
7/24/2019 Continuous random variable theory.pdf
52/101
-
7/24/2019 Continuous random variable theory.pdf
53/101
-
7/24/2019 Continuous random variable theory.pdf
54/101
We do not have explicit formula for CDF F
of X
2
. In stead values are tabulated onp. 695-696 as below (F occurs in margin here,
and related value of r.v. inside the table):
-
7/24/2019 Continuous random variable theory.pdf
55/101
If F is CDF for Chi square random variable
Having 5 degrees of freedomF(1.61)= .1 , F(4.35)= .5
P[X2 < t]
F 0.100 0.2 50 0.500
5 1.61 2 .67 4.35
6 2 .2 0 3.45 5.35
7 2 .83 4.2 5 6.35
-
7/24/2019 Continuous random variable theory.pdf
56/101
38. Consider a chi squared random variable
with 15 degrees of freedom.(i) What is the mean of ?
2
15X
Sol: Chi square distribution is Gamma with
=/2 &=2 , Mean ===15 and2 =2 = 30
-
7/24/2019 Continuous random variable theory.pdf
57/101
otherwise0
0,2)2/15(
1)(
,2/,2var
2/1)2/15(2/15
2
xexxf
hencewithiablerand omisX
x
Sol:
(ii) What is the expression for the density for
?2
15X
-
7/24/2019 Continuous random variable theory.pdf
58/101
( iii) What is the expression for the moment
Generating function for 215X
Sol:
2/1,)21()1()(2/15
ttttm x
-
7/24/2019 Continuous random variable theory.pdf
59/101
P[X2
< t]
F 0.005 0.010 0.900
15 4.60 5.2 3 2 2 .3
6 2 .2 0 3.45 5.35
7 2 .83 4.2 5 6.35
.02 5
6.2 6
10.0900.01
]3.2 2[1]3.2 2[ 215215
XPXP
( iv) Find ]3.2 2[ 215
XP
-
7/24/2019 Continuous random variable theory.pdf
60/101
875.002 5.0900.0)2 6.6()3.2 2(
]3.2 22 6.6[ 215
FF
XP
P[X2 < t]
F 0.005 0.010 0.900
15 4.60 5.2 3 2 2 .3
6 2 .2 0 3.45 5.35
7 2 .83 4.2 5 6.35
.02 5
6.2 6
]3.2 22 6.6[Find(V) 215
XP
-
7/24/2019 Continuous random variable theory.pdf
61/101
Sol. P[X2 15 >2 r]=r.
freedomofdegree150.2 5205.0
f or
P[X2 15 >2 0.05]=0.05.
P[X2 < t]
F 0.005 0.010 0.950
15 4.60 5.2 3 2 5..0
.02 5
6.2 6
V Find 2 0.05, &2 0.01 for 15
degree of freedom
-
7/24/2019 Continuous random variable theory.pdf
62/101
,6.30201.0
P[X2 15 >2 r]=r.
P[X215
>20.01
]=0.01.
P[X2 < t]
F 0.005 0.010 0.990
15 4.60 5.2 3 30.6
.02 5
6.2 6
-
7/24/2019 Continuous random variable theory.pdf
63/101
4.4 The Normal Distribution
A random variable X with density f(x)
is said to have normal distribution with
parameters and > 0, where f(x) isgiven by:
.0);,(,,2
1)(
2
2
2
xexf
x
)(0)()( fi
-
7/24/2019 Continuous random variable theory.pdf
64/101
- 1f(x)dx(ii)
),(-x0)()( xfi
-
2
1
-
2
1
2
-
2
1
2
2
2
2
2
1
2
1
let
12
1
proveto
d zed xe
d zd xzx
d xe
zx
x
-
7/24/2019 Continuous random variable theory.pdf
65/101
0
2
1
-
2
1 22
2
12
2
1d zed ze
zz
0
)(2
1
0
0
2
1
0
2
1
0
2
1
0
2
1
22
22
22
d x d ye
d yed xeII
Id zed ze
yx
yx
zz
-
7/24/2019 Continuous random variable theory.pdf
66/101
2
0
2/
0
0
)(2
12/
0
2
dd we
rd rde
w
r
0
)(2
1
0
22
d x d ye
yx
-
7/24/2019 Continuous random variable theory.pdf
67/101
2I..
ei
12
2
2
12
2
1
0
2
1
-
2
1 22
d zed ze
zz
-
7/24/2019 Continuous random variable theory.pdf
68/101
Standard Normal Distribution
If =0 =1 then normal randomvariable is called standard normal
variable symbol for normal r.v is Z.
-
7/24/2019 Continuous random variable theory.pdf
69/101
The probability density function isgiven by
).,(,2
1)(
2
2
zezf
z
d ze
d zzfzZPzF
z z
z
Z
2
2
2
1
)()()(
-
7/24/2019 Continuous random variable theory.pdf
70/101
Standard Normal distribution
Theorem: Let X be normal with meanand standard deviation. The variable
is standard normal. Z has mean 0 and
standard deviation 1.
XW
X
-
7/24/2019 Continuous random variable theory.pdf
71/101
)()()( aX
PaWPaFW
,2
1)(
2
2
2 d seaXP
a s
d zzzs ds,s,
-
7/24/2019 Continuous random variable theory.pdf
72/101
,2
1)()(
2
2
2 d seaXPaF
a s
W
d zzz
s
ds,s,
,2
1)()( 2/
2
d zeaXPaF
a
z
W
=FZ(a)
-
7/24/2019 Continuous random variable theory.pdf
73/101
Moment Generating Function
Let Z be normally distributed withparameters=0 and=1 , then the momentgenerating function for Z is given by
,
2/2)( tZ etm
-
7/24/2019 Continuous random variable theory.pdf
74/101
Rajiv
-
2
1
-
2
1
2
2
2
1
2
1
)()(
d xe
d zeeeEtm
ztz
ztztz
Z
22
22
2
2
1
22
2222
tzt
tttzzztz
-
7/24/2019 Continuous random variable theory.pdf
75/101
d wd z
d zeetm
tz
tZ
wt)-(zlet
21)(
-
22/
2
2
-
22/
2
2
21)( d weetm
w
tZ
-
7/24/2019 Continuous random variable theory.pdf
76/101
Moment Generating Function
Let X be normally distributed withparameters and, then themoment generating function for X isgiven by
, E(X2) =
2
22
)(tt
X etm
-
7/24/2019 Continuous random variable theory.pdf
77/101
)()()( )( zttxX eEeEtm
)( ztt eeE )( ztt eEe
2/
22
tt ee
-
7/24/2019 Continuous random variable theory.pdf
78/101
Mean and Standard deviation for
Normal distribution
Theorem : Let X be a normal randomvariable with parameters and. Then is the mean of X and is its standard
deviation.
-
7/24/2019 Continuous random variable theory.pdf
79/101
aF
cF ZZc)XP(a
),N(isX(i)If
d zed zzfzZPzFz zz
Z
22
2
1)()()(
)(1)(F)(Z
zFziiZ
3.5zif1)(F
-3.5zif0)(F)(
Z
Z
z
ziii
-
7/24/2019 Continuous random variable theory.pdf
80/101
-
7/24/2019 Continuous random variable theory.pdf
81/101
-
7/24/2019 Continuous random variable theory.pdf
82/101
Correction for continuity for a discrete random
variable approximated with continuous random
variable: Half Unit correction
i. P(aX b)=P(a-0.5 Xb+0.5)
ii. P(Xb)= P(Xb)= P(Xb+0.5)
= P(Xb+0.5)
iii. P(X a)= P(a-0.5 X)=P( X a-.5)
iv. P(X= a)=P (a- 0.5
-
7/24/2019 Continuous random variable theory.pdf
83/101
Normal Approximation to the
Poisson Distribution
Let X be Poisson with parameters.
Then for large values ofs, X isapproximately normal with mean
,.......3,2,1,0,!
)(
xx
sexf
xs
-
7/24/2019 Continuous random variable theory.pdf
84/101
-
7/24/2019 Continuous random variable theory.pdf
85/101
Chebyshevs Inequality
We are interested to find a lower bound
on probability that X is inside a interval
symmetric about mean or upper bound
on probability that X is outside a interval
symmetric about mean .
-
7/24/2019 Continuous random variable theory.pdf
86/101
Chebyshevs Inequality
Let X be a random variable with variance
2 and mean , then for any positivenumberm ,
P[ |X-|m ]1/m 2
Chebyshevs Inequality for upper
bound
-
7/24/2019 Continuous random variable theory.pdf
87/101
P f f Ch b h I lit
-
7/24/2019 Continuous random variable theory.pdf
88/101
Proof of Chebyshev Inequality
for upper bound for continous X
IIIIIId xxfx
d xxfxd xxfx
let
c
c
c
c
)()(
)()()()(
mc
2
222
22
-
7/24/2019 Continuous random variable theory.pdf
89/101
)0(
)()()()(
222
IIas
d xxfxd xxfx c
c
)xcrc-x-if)((
)()(
2
2
ocxas
d xxfcd xxfcc
c
-
7/24/2019 Continuous random variable theory.pdf
90/101
)()(2 cXcPcXcP
)()(2 cXcPcXcP
2
1)(
)(
mmXP
mXcP
-
7/24/2019 Continuous random variable theory.pdf
91/101
Log-Normal Distribution
The positive random variable Y is
said to have a log normal
distribution, if logeY is normally
distributed. i.e. logeYN(,)=X
Point : & are not mean & standarddeviations of Log normal randomvariable.
-
7/24/2019 Continuous random variable theory.pdf
92/101
-
7/24/2019 Continuous random variable theory.pdf
93/101
Problem 45/ p 146
Let X be normal with mean andvariance2. Let G denote thecumulative distribution for Y = eX
and let F denote the cumulativedistribution for X.
Show that (i) G(y) = F(lny),
y>0
(ii) Find density of Y
-
7/24/2019 Continuous random variable theory.pdf
94/101
G(y) = P(Yy) = P(eX y) = P(X lny)
Now X ~ N(,). Therefore,y>0
0y,(lny)F
,2
1
)ln()(
X
ln
2 2
2
d xeyXPyG
y x
.0);,(, x
1
ln2
y x
-
7/24/2019 Continuous random variable theory.pdf
95/101
0y0
0y,(lny)F
,2
1)ln()(
X
2 2
d xeyXPyG
y
0y0
0,1
dy
dF
dy
dG X
yy
2
-
7/24/2019 Continuous random variable theory.pdf
96/101
otherwise0
0y,1
2
1 22
2ln
y
eyd y
d G
,2
1
)(ln
ln
2 2
2
d xeyF
y x
Hence the density for Y is given by
-
7/24/2019 Continuous random variable theory.pdf
97/101
Hence, the density for Y is given by
.other wise0
0,);,(,
2
1)(
2
2
2
ln
ye
y
yg
y
-
7/24/2019 Continuous random variable theory.pdf
98/101
Example
From a usual pack of 52 cards, cards are
drawn randomly with replacement till the
red card appears. If X denotes the number
of card drawn, using Chebyshevsinequality, find a lower bound for P[ | X-2|
top related