2. An experimenter is shining a light through a large number, $n$, of translucent plates. The $i^{\text {th }}$ plate reduces the intensity of the light by a factor $X_{i}$. These factors $X_{i}$ are random, independent, and all uniformly distributed in the interval $(1 / 2,1)$. When the light passes through several plates, its intensity is reduced by the product of these factors. Find a constant $\mu$ such that $$ \left(\prod_{i=1}^{n} X_{i} ight)^{1 / n} \longrightarrow \mu $$ in probability as $n \longrightarrow \infty$, justifying your answer carefully (in particular, cite any theorems that you use). S.P.PB. 250