The distribution function of $\displaystyle X$ is given by

$\displaystyle F(b) = \\$

$\displaystyle 0$ if $\displaystyle b < 0 \\*$

$\displaystyle \frac{b}{4}$ if $\displaystyle 0 \leq b < 1\\$

$\displaystyle \frac{1}{2} +\frac{b-1}{4}$ if $\displaystyle 1 \leq b < 2\\$

$\displaystyle \frac{11}{12} $ if $\displaystyle 2 \leq b < 3\\$

$\displaystyle 1$ if $\displaystyle 3 \leq b\\$

I have to find

a) $\displaystyle P({X= i}), i = 1,2,3$

b) $\displaystyle P({\frac{1}{2} < X < \frac{3}{2}})$

For a) I figured possible values for the random variable $\displaystyle X$ were $\displaystyle 0,1,2,3$

Doing some subtraction I got:

$\displaystyle P(X=0) = \frac{b}{4}$

$\displaystyle P(X=1) = \frac{1}{4}$

$\displaystyle P(X=2) = \frac{11}{12} - (\frac{1}{2} + \frac{b+1}{4})$

$\displaystyle P(X=3) = 1 - \frac{11}{12} = \frac{1}{12}$

I am confused. What is b? Is it a possible value that the random variable can take on? Also, is part a) the same thing as saying "find the probability mass function"?

And for part b), I am not sure where to begin with this one. Any help is appreciated.

Thank you very much.

-Jame

(Also I apologize if the table is hard to read, that's the nicest I could get it with tex)