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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.

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#### 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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