# Singular measures or finding density distribution

I'm a little confused with the next excercise:

Let $$F$$ a probability distribution given by $$F(x)=\left\{\begin{array}{cc}0&\mbox{if}x<4\\1/5&\mbox{if }4\le x<6\\7/10&\mbox{if }6\le x<8\\23/30&\mbox{if } 8\le x<10\\1&\mbox{if }10\le x\end{array}\right.$$

Calculate $$f(6)$$.

I'm sure that $$F$$ does not have density function, but some others tell me that $$F$$ is discrete. How do you try this problem?

• $F$ is piecewise constant, so the distribution is discrete. Presumably "$f$" refers to a probability mass function (PMF) rather than a density function, and $f(6)$ is asking for the probability assigned to the value $6$. Oct 8 at 4:20
• $\displaystyle\operatorname{F}'\left(x\right) = {6 \over 5}\delta\left(x - 4\right) + {7 \over 10 }\delta\left(x - 6\right) + {23 \over 30}\delta\left(x - 8\right) + \delta\left(x - 10\right)$ Oct 8 at 5:10

$$F$$ is a cumulative distribution function - describing how probability mass accumulates with increasing $$x$$ .
Proceed under the premise that $$f$$ is intended to be that pmf .