As the title suggests, is there any named distribution whose support is $[0,1]$ or $(0,1)$ and its p.d.f. curve looks like exponential distribution (monotonically decreasing over the support)?
Probability – Exponential-Like Distribution with Support [0,1]
distributionsprobability
Best Answer
If you want a distrbution that looks identically to the exponential distribution, up to a multiplicative constant, you can use a truncated exponential distribution.
It is defined by restricting the support of an exponential distribution to the interval of interest and then re-normalizing the density to obtain a distribution. Your case would yield
$$f(x) = \frac{\lambda e^{-\lambda x}}{1 - e^{-\lambda}}$$