A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1.
X and Y are independent.
Define Z = min {X, Y}. Compute the CDF of Z ?
I really have no idea about this question.
My thought is that you need to use integration but how exactly ?
Can any experts shed some light on this one ?
If we knew the joint distribution of the random variables $X$ and $Y$, we could solve the problem. With no such information, we cannot.
We will make the simplest assumption, that $X$ and $Y$ are independent. It is likely that we are meant to assume independence, and that the condition was inadvertently left out.
We will find the cdf $F_Z(z)$ of $Z$, that is, the probability that $Z\le z$. This is clearly $0$ if $z\lt 0$. Assume now that $z\ge 0$.
We have $\min(X,Y)\gt z$ if and only if $X\gt z$ and $Y\gt z$. Suppose first that $0\le z\le 1$. Then $\Pr(X\gt z)=e^{-z}$ and $\Pr(Y\gt z)=1-z$, so for $0\le z\le 1$ we have $\Pr(Z\gt z)=e^{-z}(1-z)$, and therefore $F_Z(z)=1-e^{-z}(1-z)$.
If $z\gt 1$, then $\Pr(Z\le z)=1$, so $F_Z(z)=1$.