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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
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Grades of a Class
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Uniform Distribution
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Exponential Distribution
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Triangular Distribution
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Laplace Distribution
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Erlang Distribution
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Gamma Distribution
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Chi Square Distribution
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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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Multinomial Distribution
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Geometric Distribution
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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)
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