# Cumulative Distribution Function of a Die

Im trying to understand this question:

Find the cumulative distribution function of the outcome of a single die roll that has the number 2, 4, 6, 8, 10, and 12. Also draw a graph.

All I have came up with is that the E(x) = 7.014 p= 1/6

So would I add up all the probabilities of each outcome? Say the probability of outcome of 2 is 1/6; the outcome of 4 is 2/6; outcome of 6 is 3/6.... and so on? After i get each cumulative probability, how do i find the Cumulative Distribution Function?

• Question: Do you roll the die two times ? – callculus Oct 6 '14 at 22:48
• In this example, the die is rolled just once. – TheOneWhoIsLost Oct 6 '14 at 22:51
• Is it a 6-sided dice with the numbers 2,4,6,8,10,12 ? – callculus Oct 6 '14 at 22:53
• interesting die at any case)) – Seyhmus Güngören Oct 6 '14 at 23:01
• The cumulative probability distribution function $F_X(x)$ of a discrete random variable $X$ taking on values $x_1 < x_2 < \cdots$ with probabilities $p_1, p_2, \ldots$ respectively is a staircase function that rises from $0$ at $x=-\infty$ to $1$ at $x=\infty$ with rises of $p_i$ at $x_i$, $i = 1, 2, \ldots$. The steps themselves are of widths $x_{i+1}-x_i$, that is, $F_X(x)$ has value $p_1+p_2+\cdots+p_i$ for $x \in [x_i,x_{i+1})$. Pay attention to the $[$ and $)$ in that last sentence: they are important. In your problem, $F_X(x)=0$ for $x\in (-\infty,0)$ and $1$ for $x \in [12,\infty)$. – Dilip Sarwate Oct 6 '14 at 23:03

## 1 Answer

So would I add up all the probabilities of each outcome? Say the probability of outcome of 2 is 1/6; the outcome of 4 is 2/6; outcome of 6 is 3/6.... and so on?

The probability of $X\leq 2$ is 1/6. The probability of $X \leq 4$ is 2/6. It is always smaller or equal.

$F(x)=\begin{cases} 0, \ x < 2 \\ 1/6, \ 2\leq x <4 \\ 2/6, \ 4\leq x <6 \\ 3/6, \ 6\leq x <8 \\ 4/6, \ 8\leq x <10 \\ 5/6, \ 12\leq x <12 \\ 1, \ x \geq 12 \end{cases}$

• so the cumulutive function shows all cumulative possibilities of each outcome? – TheOneWhoIsLost Oct 6 '14 at 23:28
• $P(X\leq 4)=2/6$ means, that the probability to roll a number smaller or equal 4 is 2/6. Thus in this case it is the probability for 2 outcomes, if you do not consider the outcomes, which have the probability=0, like X=3.32. – callculus Oct 6 '14 at 23:39
• Could I say that the CDF is f(x) = x/6, if 2<= x <= 12; f(x) = 0, if x<2 or x> 12. – TheOneWhoIsLost Oct 6 '14 at 23:48
• I would take intervals, because F(3)=1/6. I made an edit. Ask, if something is not clear. – callculus Oct 7 '14 at 0:13
• Ok, that makes more sense! I appreciate you clearing that up. – TheOneWhoIsLost Oct 7 '14 at 1:00