This ppt is prepared for engineering class. Random variable and stochastic prosses and probability.

1. 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
σµ
πσ
−−
= x
X exf (3-29)
,µ
,
2
1
)(
22
2/)(
2∫∞−
−−





 −
==
x
y
X
x
GdyexF
σ
µ
πσ
σµ
(3-30)
dyexG y
x
2/2
2
1
)( −
∞−∫=
π
),( 2
σµNX
)(xfX
x
µ
Fig. 3.7
∆
)(xfX
µ ,2
σ
PILLAI

2. 2

3. 3

4. 4

5. 5

6. 6

7. 7

8. 8

9. 9
Grades of a Class

10. 10
Uniform Distribution

11. 11

12. 12

13. 13
Exponential Distribution

14. 14

15. 15
Triangular Distribution

16. 16
Laplace Distribution

17. 17
Erlang Distribution

18. 18
Gamma Distribution

19. 19

20. 20
Chi Square Distribution

21. 21

22. 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,)( nkqp
k
n
kXP knk
=





== −
(3-44)
,)(λPX
.,,2,1,0,
!
)( ∞=== −
k
k
ekXP
k
λλ
(3-45)
k
)( kXP =
Fig. 3.17
12 n
)( kXP =
Fig. 3.18 PILLAI

23. 23

24. 24

25. 25
Multinomial Distribution

26. 26

27. 27
Geometric Distribution

28. 28
4. Hypergeometric:
5. Geometric: ∼ if
6. Negative Binomial: ~ if
7. Discrete-Uniform:
We conclude this lecture with a general distribution due
PILLAI
(3-49)
(3-48)
(3-47)
.,,2,1,
1
)( Nk
N
kXP ===
),,( prNBX
1
( ) , , 1, .
1
r k r
k
P X k p q k r r
r
−
− 
= = = + − 

.1,,,2,1,0,)( pqkpqkXP k
−=∞=== 
)( pgX
, max(0, ) min( , )( )
m N m
k n k
N
n
m n N k m nP X k
  
  
   
   
 
 
 
 
−
−
+ − ≤ ≤= = (3-46)

29. 29

30. 30

31. 31