Probability theory and application 2.pdf

Probability theory

1. MA208 –Probability Theory and
Applications
ONE-DIMENSIONALRANDOMVARIABLES
Dr. Kedarnath (MACS) February 3, 2021

2. ONEDIMENSIONALRANDOMVARIABLES
EVENTS DEFINED BY RANDOMVARIABLE:
LetXbe a Random Variable andx2Ris xed. We can dene the
following events byX:
f!2jX(!) =xg= (X=x)
f!2jX(!)xg= (Xx)
f!2jX(!)xg= (Xx)
f!2jx1X(!)x2g= (x1Xx2)
Dr. Kedarnath (MACS) February 3, 2021

3. EXAMPLE
=fHHH; HHT; HT H; HT T; T HH; T HT; T T H; T T Tg
jj= 2
3
= 8
(i)A=f!2jX(!) = 2g= (X= 2) =fHHT; HT H; T HHg
P(X= 2) =
3
8
(ii)B=f!2jX(!)<2g= (X <2) =fHT T; T HT; T T H; T T Tg
P(B) =P(X <2) =
4
8
=
1
2
TYPES OFRANDOMVARIABLES
1)
2)
3)
Dr. Kedarnath (MACS) February 3, 2021

4. DISCRETERANDOMVARIABLE(DRV)
LetXbe a Random Variable.Xis DRV ifjRXjis nite or countable.
Note:RX=fx1; x2; : : : ; xn; : : :g
CONTINUOUSRANDOMVARIABLE(CRV)
IfjRXjis uncountable,Xis called CRV.
PROBABILITYMASSFUNCTION(PMF)OFDRV
LetXbe DRV andRX=fx1; x2; : : :g:A functionp
x
:R![0;1]dened by
p
x
(xi) =P(X=xi)is called PMF ofXif the following hold
(i)p
x
(xi)0for allxi2RX
(ii)
1
X
i=1
p
x
(xi) = 1
Note: The collection of pairs(xi;p
x
(xi)); i= 1;2: : :is called Probability Distribution of
the RVX:
Dr. Kedarnath (MACS) February 3, 2021

5. Dr. Kedarnath (MACS) February 3, 2021

6. EXAMPLE
=f1;2;3;4;5;6g
DeneX(!) =!; !2
RX=f1;2;3;4;5;6g
PMFp
x
(xi) =?
p
x
(k) =P(X=k); k= 1;2;3;4;5;6
So,p
x
(1) =p
x
(2) =p
x
(3) =p
x
(4) =p
x
(5) =p
x
(6) =
1
6
Dr. Kedarnath (MACS) February 3, 2021

7. BINOMIALDISTRIBUTION
Consider an random experiment of tossing a fair coin forntimes.
=fT T T
|{z}
n fold
; HT T HHT; T
.
.
.
T H T HHT T
.
.
.
.
.
.
.
.
.
T T T H
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
jj= 2
n
; n2N
DeneX: !RbyX(!) = #H's in!; !2:
Also dene
P(occurance of aH)=p;0p1
P(occurance of aT)= 1p:
ThenXis a Binomial Random Variable with PMF.
p
x
(k) =P(X=k)
=

n
k
!
p
k
(1p)
nk
;wherek= 0;1;2; ; n
Dr. Kedarnath (MACS) February 3, 2021

8. Note: A DRVXis a Binomial Random Variable with parameters(n;p)
if the PMF ofXis given by
p
x
(k) =P(X=k) =

n
k

p
k
(1p)
nk
;wherek= 0;1;2; ; n
EXERCISE:
Check that, ifXis Binomial Random Variable with parameters(n;p)
(XBi(n;p));
n
X
k=0
p
x
(k) =
n
X
k=0

n
k

p
k
(1p)
nk
= 1
(Use Binomial Theorem)
Dr. Kedarnath (MACS) February 3, 2021

9. Dr. Kedarnath (MACS) February 3, 2021

10. DEFINITION(BINOMIALDISTRIBUTION)
A DRVXis called Binomial distribution with parameters(n;p)if the
PMF ofXis given by
p
x
(k) =P(X=k) =

n
k

p
k
(1p)
nk
;wherek= 0;1;2; ; n
Dr. Kedarnath (MACS) February 3, 2021

11. EXERCISE(RELATED TOPMF)
(a) p(x)dened by
p(x) =
8
>
<
>
:
3
4

1
4

x
x= 0;1;2
0 otherwise
is a PMF of a discrete RVX:
(b)
(i)P(X= 2)
(ii)P(X2)
(iii)P(X1)
Dr. Kedarnath (MACS) February 3, 2021

12. Dr. Kedarnath (MACS) February 3, 2021