The cumulative distribution function of a discrete random variable $X$ is given by $$F_X(t) = \begin{cases}0, & t < 0, \\ \frac13, & 0\le t < \frac12, \\ a, & \frac12 \le t < \frac34, \\ 1, & t\ge \frac34.\\ \end{cases}$$ Determine the constant $a$, if $\mathbb P\left(X > \frac58\right) = \frac 12$ and the probability mass function of $X$.
so in order to find the value of $a$ we will add all the values of $p(x)$ and equal it to $1$. So, in this case, it should be like $p(0)+p(1/2)+p(5/8)=1$ But the answer is coming wrong. So why can't we solve it like this? and what is the correct method otherwise??
The equation $\sum \mathbb P_X(x)=1$ give you nothing!(except $\frac{1}{3}\leq a \leq 1$)
$$1=\mathbb P_X(0) + \mathbb P_X(\frac{1}{2})+\mathbb P_X(\frac{3}{4}) =\frac{1}{3}+(a-\frac{1}{3})+(1-a)=1$$.
You should use information of $\mathbb P(X >\frac{5}{8})=\frac{1}{2}$
$$\mathbb P(X >\frac{5}{8})=\frac{1}{2}=1-\mathbb P(X \leq \frac{5}{8})$$
so since $\frac{1}{2} <\frac{5}{8}<\frac{3}{4}$
$$\mathbb P(X \leq \frac{5}{8})=\frac{1}{2}=\mathbb F_X(\frac{5}{8})=a$$