Question

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the cumulative distribution function

Answer 1

Let X be the random variable denotes the total score in two thrown of a die.

Sample space S

I\II	1	3	3	5	5	5
1	2	4	4	6	6	6
3	4	6	6	8	8	8
3	4	6	6	8	8	8
5	6	8	8	10	10	10
5	6	8	8	10	10	10
5	6	8	8	10	10	10

n(S) = 36

X = {2, 4, 6, 8, 10}

Values of the random variable	2	4	6	8	10	Total
Number of elements in inverse image	1	4	10	12	9	36

Cumulative distribution function
F(x) = P(X ≤ x)

= `sum_(x_"i" ≤ x) "P"("X" = x_"i")` 

F(2) = P(X < 2)

= P(X < 2) + P(X = 2)

= `0 + 1/36`

= `1/36`

F(4) = `"P"("X" ≤ 4)`

=  P(X <2) + P(X = 2) + P(X = 4)

= `0+ 1/36 + 4/36`

= `5/36`

F(6) = `"P"("X" ≤ 6)`

= P(X < 2) + P(X = 2) + P(X = 4) + P(X = 6)

= `0 + 11/36+ 4/36 + 10/36`

= `15/36`

F(8) = P(X ≤ 8)

= P(X < 2) + P(X = 2) + P(X = 4) + P(X = 6) + P(X = 8)

= `0 + 1/6 + 4/36 + 10/36 + 12/36`

= `27/36`

F(10) = P(X ≤ 10)

= P(X < 2) + P(X = 2) + P(x = 8) + P(X = 10)

= `0 + 1/36 + 4/36 + 10/36 + 12/36 + 9/36`

= `36/36`

= 1

F(x) = `{{:(0",",  "For"  - oo < x < 2),(1/36",",  "For"  2 ≤ x ≤ 4),(5/36",",  "For"  4 ≤ x < 6),(15/36",",  "For"  6 ≤ x < 8),(27/36",",  "For"  8 ≤ x < 10),(1",",  "For"  10 ≤ x < oo):}`