I want to compute the quartiles of X ∼ Γ(1,β)? Let X ∼ Γ(1,β). Find the first, second, and third quartiles of X, that is, the values xp for which P(X < x.25) = .25, P(X < x.50) = .5, and P(X < x.75) = .75.
I want to compute the quartiles but i am not getting how to do it given  probability ?
 A: Hint: if $X\sim\Gamma(1,\beta)$, then $X\sim\exp(\beta)$, for $\beta>0$, so
$$
F(t) = P(X\le t) = 1 -  e^{-\beta t}, \quad \text{for } t>0.
$$
The above is called the (cumulative) distribution function (as mentioned by Daniel in the comments). The quantile function is the inverse of the distribution function. 
For example, say you want to find $x_{0.50}$, then you must solve the following equation for $t$:
$$
F(t) = 0.50,
$$
in this case (the exponential distribution), the quantile function has a nice closed form. Can you proceed from here?
A: This appears to be a gamma distribution with shape parameter 1 and rate parameter $\beta,$ which is also an exponential distribution with rate parameter $\beta.$
You can look at Wikipedia 'gamma distribution' and 'exponential distribution'.
The latter has the CDF. From there, you should be able to find the upper and lower quartiles and the median.
You may want to compare formulas shown there with ones in your textbook, to
make sure you understand the notation. (@ K. Brix (+1) has given you a good start.)
From R statistical software with $\beta = 1,$ the results are:
> qexp(c(.25, .5, .75), rate=1)
## 0.2876821 0.6931472 1.3862944
> qgamma(c(.25, .5, .75), 1, 1)
## 0.2876821 0.6931472 1.3862944

The figure below shows the PDF (left) and CDF for $\beta=1.$
Under the PDF curve, each of the regions separated by vertical red lines
has area (probability) 1/4. 

