Probability best notes

15. Probobility Distribution

Ex. (1). A rqndom voriqble X hqs the following probobility distribution :




P(X : x)
Find(i)k(ii)p(x<3)(iii)p(x>2)(iv)P(0<x<a)(v)P(2<x<5)
Solution : For o rqndom vorioble Xwehove fi=l
o,='
:. k+3k+ 5k+7k+9k+ llk+l3k:l
i.e.49k:13 k:L49

P(X : x)


(i) k: qs
(iD P(X < 3) : P(X : 0) + P(x :1) + P(X:2)
-49 49 49 49
(iii)P(x22):P(X:2)+P(X:3)+P(X:4)+P(X:5)+P(X:6)
(iv)P(0 < X < 4) : P(X: 1) + P(X : 2) + P(X : 3)
(v) P(2 < X <5) : P(X : 2) +P(X : 3) + P(X : 4) + P(X : 5)
s 7 9 ll 32
:49+ *+ 49* qg:49
Ex. (2). Colculqte the Expected volue ond vorionce of x if x denotes the
number obtoined on the uppermost fqce when o fqir die is thrown.
:
Solution : When o foir die is thrown, the somple spoce is ^S {1,2, 3, 4, 5, 6\ '
Let X denotes the number obtoined on the uppeflnost foce.
.'. Xcqn toke volues l, 2, 3, 4, 5', 6'
: : : :
P(X : l) : P(X : 2) : P(X : 3) : P(X 4) P(X 5) P(X 6)
:
I
6
The probobilitY distribution is




ffim'
Aurangabad)
'
Satish Jadhwar ( Sant Meera Jr' College'

, x.2 .D
rt
l




(i) Expected Volue : E(X) : i*,, p, =1= 3.5

(ii) Vorionce : V(n: E(X,) - lE(X)],


**i,0,-(2.,,0,)'
: T_a'=?_+
182-r47
_
t2
.'. Vqriqnce : V(X): :2.9767
i
Bx. (3). A discrete rondom voriqble Xtqkes the volues -1, 0 snd,2 with the
111
probobilities 4' 2' 4 respectively. Find Y($ ondstqndqrd Deviotion.
solution : Given thqt the rqndom vorioblextokes the volues -1, 0 snd2.
The corresponding probobilities qre
i,;,i
p(-l) : i,p(o): j p@:
ona
Given doto cqn be tobuloted qs follows
I




(i) Vorionce: V(X): E(X) - lE(X)],




I, 0ggl
Ex. @) The p. d. f. ofX, find p(X < 1) ond p(lxl< 1) where

f@)=!J2
l8
if-2<x<4
:0 otherwise.




Satish Jadhwar ( Sant Meera Jr. College, Aurangabad)