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Question
A random variable X has the following probability distribution :
| X=x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| P[X=x] | k | 3k | 5k | 7k | 9k | 11k | 13k |
(a) Find k
(b) find P(O <X< 4)
(c) Obtain cumulative distribution function (c. d. f.) of X.
Solution
sum P(x)=1
P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=1
K+3x +5k +7k +9k +11k +13k=1
49k=1
k=1/49
P(0<x<4)=P(1)+P(2)+p(3)
=3k+5k+7k
=15k
=15/49
`F(0)=P(0)=1/49`
`F(1)=P(0)+P(1)=1/49+3/49=4/49`
`F(2)=P(0)+P(1)+P(2)=1/49+3/49+5/49=9/49`
`F(3)=P(0)+P(1)+P(2)+P(3)=1/49+3/49+5/49+7/49=16/49`
`F(4)=P(0)+P(1)+P(2)+P(3)+P(4)=1/49+3/49+5/49+7/49+9/49=25/49`
`F(5)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)=1/49+3/49+5/49+7/49+9/49+11/49=36/49`
`F(6)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=1/49+3/49+5/49+7/49+9/49+11/49+13/49=49/49`
Cummulative districbution function (c.d.f.) of x
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| F(x) | 1/49 | 4/49 | 9/49 | 16/46 | 25/49 | 36/49 | 1 |
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