II. 1. The range of a random variable X is {0,1,2). Given that
P(X = 0) = 3c", P(X = 1)=4c-10c2, P(X = 2)=5c-1
(i) Find the value of c (ii) P(X<I), P(1<x2) and (P(0<X53)
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The range of a random variable X is {0,1,2}.
Given that P(X=0)=3C
3
,P(X=1)=4C−10C
2
,P(X=2)=5C−1. Find the value of:
(i) C
(ii) P(X<1)
(iii) P(1<X≤2) and P(0<X≤3).
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Solution To Question ID 609962
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Answer
(i) We must have ∑P(X)=1
⇒P(X=0)+P(X=1)+P(X=2)=1
⇒(3C
3
)+(4C−10C
2
)+(5C−1)=1
⇒3C
3
−10C
2
+9C−2=0
⇒(C−1)(3C
2
−7C+2)=0
⇒(C−1)(C−2)(3C−1)=0
⇒C=1,2,
3
1
Also 0≤P(X=0),P(X=1),P(X=2)≤1
So only acceptable value of C is
3
1
(ii) P(X<1)=P(X=0)=3(
3
1
)
3
=
9
1
(iii) P(1<X≤2)=P(X=2)=4C−10C
2
=4⋅
3
1
−10⋅
9
1
=
9
2
and
P(0<X≤3)=1−P(X=0)=1−
9
1
=
9
8
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