When solving for percentiles with cdf, Should I discard constants?
From my book: $F(x_p) = P(X \le x_p) = \int_{-\infty}^{x_p} f(x) dx = \frac{p}{100}$
If I don't have the cdf formula, I can integrate the pdf $f(x)$ for $F(x)$, but there's the missing constant that the $F(x)$ might have had.
Example case:
cdf: $ F(x) = -\frac{4}{3x} + \frac{4}{3} $
pdf: $ f(x) = \frac{d}{dx} F(x) = \frac{4}{3x^2} $
Solving for median (50th percentile) using $F(x_m) = 0.5$:
$ F(x_m) = 0.5 = -\frac{4}{3x_m} + \frac{4}{3} $
$x_m = \frac{8}{5} $
Solving for median (50th percentile) using $\int_{-\infty}^{x_m} f(x) dx = 0.5$:
$ \int_{-\infty}^{x_m} f(x) dx = 0.5 = -\frac{4}{3x} |^{x_m} + C = -\frac{4}{3x_m} + C $
$ x_m = -\frac{8}{3} $
The answers I got are radically different, so either I am doing my math wrong, or the constant is shifting the answer a lot.