Let X be a random variable with the following Cumulative Distribution Function ([x] denotes the greatest integer function less than or equal to x)
$F_X(x)$ = $ \left \{ \begin{array} {cc} 0 & x<1 \\ c & 1 \leq x < 2 \\ c+ \sum_{j=1}^{[x]-1} (\frac {3}{10})^j & otherwise \end{array} \right . $
Find the value of constant, c.
I have considered using the fact that $ \lim_{x \rightarrow \infty} F_x(x) =1$ But I am confused whether we add all the $F_X$? Also, if we do what will happen when put limit x tending to infinity in the greatest integer function? Can someone help me with this?
Since $F_X$ is a CDF and $\sum_{j=1}^{\infty}\left(\frac3{10}\right)^j=\frac37$ we may conclude that $$1=\lim_{x\to\infty}F_X(x)=c+\frac3{7}$$ So: $$c=\frac47$$