1 Continuous-type random variables 1. Normal (Gaussian): X is said to be normal or Gaussian r.v, if This is a bell shaped curve, symmetric around the parameter and its distribution function is given by where is often tabulated. Since depends on two parameters and the notation will be used to represent (3-29). . 2 1 ) ( 2 2 2 / ) ( 2 x X e x f (3- 29) , , 2 1 ) ( 2 2 2 / ) ( 2 x y X x G dy e x F (3- 30) dy e x G y x 2 / 2 2 1 ) ( ) , ( 2 N X ) ( x f X x Fig. 3.7 ) ( x f X , 2 PILLAI
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1
Continuous-type random variables
1. Normal (Gaussian): X is said to be normal or Gaussian r.v, if
This is a bell shaped curve, symmetric around the parameter and its distribution function is given by
where is often tabulated. Since depends on two parameters and the notation will be used to represent (3-29).
.2
1)(
22 2/)(
2
x
X exf (3-29)
,
,2
1)(
22 2/)(
2
x y
X
xGdyexF
(3-30)
dyexG yx 2/2
2
1)(
),( 2NX)(xf X
xFig. 3.7
)(xf X
,2
PILLAI
2
3
4
5
6
7
8
9
Grades of a Class
10
Uniform Distribution
11
12
13
Exponential Distribution
14
15
Triangular Distribution
16
Laplace Distribution
17
Erlang Distribution
18
Gamma Distribution
19
20
Chi Square Distribution
21
22
Discrete-type random variables
1. Bernoulli: X takes the values (0,1), and
2. Binomial: if (Fig. 3.17)
3. Poisson: if (Fig. 3.18)
.)1( ,)0( pXPqXP (3-43)
),,( pnBX
.,,2,1,0 ,)( nkqpk
nkXP knk
(3-44)
, )( PX
.,,2,1,0 ,!
)( kk
ekXPk
(3-45)
k
)( kXP
Fig. 3.17
12 n
)( kXP
Fig. 3.18 PILLAI
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24
25
Multinomial Distribution
26
27
Geometric Distribution
28
4. Hypergeometric:
5. Geometric: if
6. Negative Binomial: ~ if
7. Discrete-Uniform:
We conclude this lecture with a general distribution duePILLAI
(3-49)
(3-48)
(3-47)
.,,2,1 ,1
)( NkN
kXP
),,( prNBX1
( ) , , 1, .1
r k rkP X k p q k r r
r
.1 ,,,2,1,0 ,)( pqkpqkXP k
)( pgX
, max(0, ) min( , )( )
m N m
k n kN
n
m n N k m nP X k
(3-46)
29
30
31
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