Order Statistics

Suppose that $X$, $Y$, and $Z$ have a joint density function given by

Compute $\mathbb {P}\left(X < Y < Z\right)$ and $\mathbb {P}\left(X = Y < Z\right)$.

Two points are chosen uniformly and independently on the perimeter of a circle of radius $1$. This divides the perimeter into two pieces. Determine the expected value of the length of the shorter piece.

Let $X_1$ and $X_2$ be independent, $U(0,1)$-distributed random variables, and let $Y$ denote the point that is closest to an endpoint. Determine the distribution of $Y$.

The statistician Piggy has to wait an amount of time $T_0$ at the post office on an occasion when she is in a great hurry. In order to investigate whether or not chance makes her wait particularly long when she is in a hurry, she checks how many visits she makes to the post office until she has to wait longer than the first time. Formally, let $T_1, T_2, \ldots$ be the successive waiting times and $N$ be the number of times until some $T_k > T_0$, that is,

What is the distribution of $N$ under the assumption that $\left\{ T_n, n \geq 0\right\}$ are i.i.d. continuous random variables? What can be said about $\mathbb {E}\left[N\right]$?

Let $X_1, X_2, \ldots , X_n$ be independent, continuous random variables with common distribution function $F(x)$, and consider the order statistic $(X_{(1)}, X_{(2)}, \ldots , X_{(n)})$. Compute $\mathbb {E}\left[F(X_{(n)}) - F(X_{(1)})\right]$.

Let $X_1$, $X_2$, $X_3$, and $X_4$ be independent, $U(0,1)$-distributed random variables. Compute

Let $X_1$, $X_2$, $X_3$ be independent $U(0,1)$-distributed random variables, and let $X_{(1)}, X_{(2)}, X_{(3)}$ be the corresponding order variables. Compute $\mathbb {P}\left(X_{(1)} + X_{(3)} \leq 1\right)$.

Suppose that $X_1, X_2, X_3, X_4$ are independent $U(0,1)$-distributed random variables and let $(X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)})$ be the corresponding order statistic. Compute

Suppose that $X_1, X_2, X_3, X_4$ are independent $U(0,1)$-distributed random variables and let $(X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)})$ be the corresponding order statistic. Find the distribution of

Suppose that $X_1, X_2, X_3$ are independent $U(0,1)$-distributed random variables and let $(X_{(1)}, X_{(2)}, X_{(3)})$ be the corresponding order statistic. Compute $\mathbb {P}\left(X_{(1)} + X_{(2)} > X_{(3)}\right)$.

Suppose that $X_1, X_2, X_3$ are independent $U(0,1)$-distributed random variables and let $(X_{(1)}, X_{(2)}, X_{(3)})$ be the corresponding order statistic. It is of course a trivial observation that we always have $X_{(3)} \geq X_{(1)}$. However,

Let $X_1, X_2, X_3$ be independent $U(0,1)$-distributed random variables, and let $X_{(1)}, X_{(2)}, X_{(3)}$ be the ordered sample. Let $0 \leq a < b \leq 1$. Compute

Let $X_1, X_2, \ldots , X_8$ be independent $Exp(1)$-distributed random variables with order statistic $(X_{(1)}, X_{(2)}, \ldots , X_{(8)})$. Find

Let $X_1, X_2, X_3$ be independent $U(0,1)$-distributed random variables, and let $X_{(1)}, X_{(2)}, X_{(3)}$ be the order statistic. Prove the intuitively reasonable result that $X_{(1)}$ and $X_{(3)}$ are conditionally independent given $X_{(2)}$ and determine this (conditional) distribution.

The random variables $X_1$, $X_2$, and $X_3$ are independent and $Exp(1)$-distributed. Compute the correlation coefficient $\rho_{X_{(1)}, X_{(3)}}$.

Let $X_1$ and $X_2$ be independent, $Exp(a)$-distributed random variables.

1. Show that $X_{(1)}$ and $X_{(2)} - X_{(1)}$ are independent, and determine their distributions.

2. Compute $\mathbb {E}\left[X_{(2)} \mid X_{(1)} = y\right]$ and $\mathbb {E}\left[X_{(1)} \mid X_{(2)} = x\right]$.

Let $X_1$, $X_2$, and $X_3$ be independent, $U(0,1)$-distributed random variables. Compute $\mathbb {P}\left(X_{(3)} > 1/2 \mid X_{(1)} = x\right)$.

Suppose that $X \in U(0,1)$. Let $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$ be the order variables corresponding to a sample of $n$ independent observations of $X$, and set

Show that

The random variables $X_1, X_2, \ldots , X_n, Y_1, Y_2, \ldots , Y_n$ are independent and $U(0,a)$-distributed. Determine the distribution of

Let $X_1, X_2, \ldots , X_n$ be independent, $Exp(a)$-distributed random variables, and set

The purpose of this problem is to provide a probabilistic proof of the relation

Let $X_1, X_2, \ldots , X_n$ be independent, $Exp(1)$-distributed random variables. Consider the usual order variables $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$, and set

Let $X_1, X_2, \ldots , X_n$ be independent, $Exp(1)$-distributed random variables, and set

Compute $\mathbb {E}\left[Z_n\right]$ and $\operatorname {Var}\left[Z_n\right]$.

The random variables $X_1, X_2, \ldots , X_n$ are independent and $Exp(1)$-distributed. Set

Show that $V_n \overset {d}{=} W_n$.

Let $X_1, X_2, \ldots , X_n$ be independent, $Exp(a)$-distributed random variables. Determine the distribution of $\sum_{k=1}^{n} X_{(k)}$.

Let $X_1, X_2, \ldots , X_n$ be i.i.d. random variables and let $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$ be the order variables. Determine

The number of individuals $N$ in a tribe is $Fs(p)$-distributed. The lifetimes of the individuals in the tribe are independent, $Exp(1/a)$-distributed random variables, which, further, are independent of $N$. Determine the distribution of the shortest lifetime.

Let $X_1, X_2, \ldots$ be independent, $U(0,1)$-distributed random variables, and let $N \in Po(\lambda )$ be independent of $X_1, X_2, \ldots$. Set

($V = 0$ when $N = 0$). Determine the distribution of $V$, and compute $\mathbb {E}\left[V\right]$.

Let $X_1, X_2, \ldots$ be $Exp(\theta )$-distributed random variables, let $N \in Po(\lambda )$, and suppose that all random variables are independent. Set

Show that $Y \overset {d}{=} \max \left\{ 0, V\right\}$, where $V$ has a Gumbel type distribution.

Suppose that the random variables $X_1, X_2, \ldots$ are independent with common distribution function $F(x)$. Suppose, further, that $N$ is a positive, integer-valued random variable with generating function $g(t)$. Finally, suppose that $N$ and $X_1, X_2, \ldots$ are independent. Set

Show that