Why is $P(X=2.5)=0?$

Question

The discrete random variable X has cdf

$$\left\{ \begin{array}{rcr} 0 & = & \text{for} \ x<1 \\ 1/4 & = & \text{for} \ 1\le x<2 \\ 3/4 & = & \text{for} \ 2\le x<3 \\ 1 & = & \text{for} \ x\ge 3 \end{array} \right.$$

Find a) $P(X=1)$, b) $P(X=2)$, c) $P(X=2.5)$, d) $P(x\leq2.5).$

I found a), b) and d) easy by drawing out the peacewise cdf. I fail to se why $P(X=2.5)=0$. If $2.5\in 2\le x<3,$ shouldn't the answer be the same as in b), that is $\frac{3}{4}-\frac{1}{4} = \frac{1}{2}?$

Answer 1

If you draw the cumulative distribution function you will find that the graph has 'jumps', i.e. is discontinuous at the points $x=1,2,3$. Notice that we are working with a discrete random variable, hence $X$ can take on the values $1,2$ or $3$, with respective probabilities $\frac{1}{4},\frac{1}{2}$ and $\frac{1}{4}$. You can see this in your c.d.f. by looking at the 'size' of the jump at the given points.

Since $X$ can only take on values $1,2$ and $3$ (it is discrete) then $P(X=2.5)=0$.

Answer 2

It might help to express this as a probability mass function.

$$P_X(x)=\left\{ \begin{array}{rcr} 1/4 & x=1 \\ 2/4 & x=2 \\ 1/4 & x=3 \\ 0 & \text{otherwise} \end{array} \right.$$