I'm trying to find the conditional probability density function for the following scenarios and wondering if they are correct
A number x is chosen at random in the interval [0, 1], given that x > 1/4.
$f(x|1/4<x<1) = 4/3 \space for \space 1/4<x<1$
$f(x|1/4<x<1) = 0 \space otherwise$
A number t is chosen at random in the interval [0, $\inf$) with exponential density $e^t$, given that 1 < t < 10.
$f(t|1<t<10) = \frac{e^{-t}}{e^{-1}-e^{-10}} \space for \space 1<t<10$
$f(t|1<t<10) = 0 \space otherwise$
A dart is thrown at a circular target of radius 10 inches, given that it falls in the upper half of the target.
$f((x,y)|y>0) = \frac{1}{50\pi} \space for \space y>0 \space and \space x^2+y^2<100$
$f((x,y)|y>0) = 0 \space otherwise$
Two numbers x and y are chosen at random in the interval [0, 1], given that x > y.
$f((x,y)|x > y) = 2 \space for \space x > y \space and \space x,y\in[0,1]$
$f((x,y)|x > y) = 0 \space otherwise$
Correct.
Well, that is assuming that darts do indeed land on dart boards with uniform distribution.
The support for the last one can also be expressed as $0\leqslant y<x\leqslant 1$.