How can I calculate the variance of a minimum of two random variables with different pdf and cdf? I am trying to solve a problem where I need to find the variance of min (a,b). a is actually a function of a uniformly distributed r.v. while b is another r.v. with pdf and cdf as f and F. The support for both is [0,inf]. Any suggestions please?
Thanks in advance.
If $a$ is a function of a uniformly distributed r.v. then it is just another r.v.

Are your r.v.s indepentent?

I think it should work to write everything in terms of indicator functions:

$c = \min(a,b) = a\cdot 1_{\{ a&lt;b \}} + b \cdot 1_{\{ b\leq a \}}
= a\int_a^\infty f(x) dx + b\int_0^a f(x) dx$.

Now you have to find the variance of the sum. Since your r.v.s live $[0,\infty]$, I would suggest computing the Laplace transform $\mathbb{E}[\exp(-sc)]$ (note I named $c = \min(a,b)$). Of course, the outcome depends on $a$ and $f$. If you have this, you can compute the first and second moment (which you need for the variance) by differentiation w.r.t. $s$.

Hope this helps.
If $a$ is a function of a uniformly distributed r.v. then it is just another r.v.

Are your r.v.s indepentent?

I think it should work to write everything in terms of indicator functions:

$c = \min(a,b) = a\cdot 1_{\{ a&lt;b \}} + b \cdot 1_{\{ b\leq a \}}
= a\int_a^\infty f(x) dx + b\int_0^a f(x) dx$.

Now you have to find the variance of the sum. Since your r.v.s live $[0,\infty]$, I would suggest computing the Laplace transform $\mathbb{E}[\exp(-sc)]$ (note I named $c = \min(a,b)$). Of course, the outcome depends on $a$ and $f$. If you have this, you can compute the first and second moment (which you need for the variance) by differentiation w.r.t. $s$.

Hope this helps.