Given a CDF, find P(-.5<X<.5)

Question

Given the following CDF: \begin{equation*} F(x)= \left\{ \begin{array}{lr} 0 & x<-1, \\ \frac{x+2}{4} & -1 \leq x < 1 \\ 1 & x \leq 1 \end{array} \right. \end{equation*} Compute P(-.5 < X<.5)

I graphed out the CDF. I just wanted to know if the way to go about this was integrating (x+2)/4 over -1 to 1 or if there was a different way to go about about this. I'm not sure if my method is 100% correct.

Thanks all for your support.

Answer 1

You integrate the PDF, not the CDF, to obtain probabilities. Also it doesn't make sense that you would integrate from $-1$ to $1$ since the probability you're asked to find is that of $[-0.5<X<0.5]$.

To find $P(a<X<b)$ for $a<b$ you need to argue a little differently. Recall, that if $F$ is the CDF of $X$, then $F(x)=P(X\leq x)$ for all $x$. Now use that $$ P(a<X<b)=P(X<b)-P(X\leq a) $$ since $[a<X<b]=[X<b]\setminus [X\leq a]$. To find the first probability, use that $$ P(X<b)=F(b-)=\lim_{x\,\uparrow\, b}F(x). $$

Answer 2

If the CDF is given then just read off the values from it without having to integrate.

$$\begin{align}P(-0.5<X<0.5)&=F(0.5)-F(-0.5)\\ &=\frac {2.5}4-\frac{1.5}4 \\&=0.25\end{align}$$