Finding a CDF given a PDF using summations

Question

I am in a prob and stats class and we have just begun our discussion on discrete random variables. I am given a pdf of

$$ f(x) = \left\{\begin{aligned} &x/10 &&: x = 1,2,\ldots,4\\ &0 &&: \text{otherwise} \end{aligned} \right.$$

I need to find an expression for the cumulative distributive function. I have done some research and I see that you can do this with derivatives and anti-derivatives.. however, that is not how the examples my professor has done are shown so I am looking for help with that. I have the following definition:

$$F(x) = \sum_{x_i \le x} f(x_i)$$ where the summation is taken over indices $i$ such that $ x_i \le x$. I have the list of properties of CDF I am just not sure how to apply it. For my particular problem I think I need to do

$$F(X) = \sum_{k=1}^x \frac k {10}$$ which would be $$ \frac 1 {10} \sum_{k=1}^x k$$

This looks correct. You don't need derivatives for discrete random variables; they are needed for continuous random variables. — angryavian, Jun 11 '15 at 14:59

score 0 · Answer 1 · answered Jun 11 '15 at 14:58

0

Yes you are correct. For discrete random variables, you should sum the lesser values as you have defined.

Derivatives and anti-derivatives are used for continuous random variables. You'll probably cover this topic in your class soon.

answered Jun 11 '15 at 14:58

James

989

so my summation would $\frac {x(x +1)} {20}$ – Jun 11 '15 at 15:12

Finding a CDF given a PDF using summations

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