Give the definition of an exponential random variable $X$ with parameter $\lambda$. Show that $X$ is memoryless.

Now let $X, Y$ be independent exponential random variables, each with parameter $\lambda$. Find the probability density function of the random variable $Z=\min (X, Y)$ and the probability $P(X>Y)$.

Suppose the random variables $G_{1}, G_{2}$ are independent and each has probability density function given by

$$f(y)=C^{-1} e^{-y} y^{-1 / 2}, \quad y>0, \quad \text { where } C=\int_{0}^{\infty} e^{-y} y^{-1 / 2} d y$$

Find the probability density function of $G_{1}+G_{2} \cdot$ [You may use standard results without proof provided they are clearly stated.]