Q3
Continuous Probability Distributions and Random Variables
Distribution of Transformed or Combined Random Variables
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Consider a region $R$ defined by $0 < x < 1$ and $0 < y < 1$ in the $x y$-plane. We randomly select a point on $R$ and refer to the selected point as A. We assume that A is uniformly distributed on $R$. Let AB be a perpendicular line from A to the $y$-axis and AC be a perpendicular line from $A$ to the $x$-axis as shown in the figure. We call rectangle $OCAB$ as "the rectangle of A", where O denotes the origin. Let $S$ be a random variable representing the area of the rectangle of A. Answer the following questions.
(1) Calculate the expectation value of $S$.
(2) Calculate the probability that $S \leq r$ holds, where $0 < r < 1$.
(3) Calculate the probability density function of $S$.
Again consider the region $R$. Let $n$ be a positive integer. We select $n$ points on $R$ and refer to the selected points as $\mathrm{A}_1, \mathrm{~A}_2, \ldots, \mathrm{~A}_n$. We assume that each of the points is uniformly distributed on $R$, and $\mathrm{A}_i$ and $\mathrm{A}_j$ for $i \neq j$ are selected independently. Answer the following question.
(4) Let $S_i$ be a random variable representing the area of the rectangle of $\mathrm{A}_i$. Let $Z$ be a random variable which is the minimum of $S_1, S_2, \ldots, S_n$. Calculate the probability density function of $Z$.