Articles of order statistics

Variance of event counting

I have this question (not homework, review problem for qualifying exam), tried approaching it a couple of ways (unsuccessfully). Any recommendations? Let $X_1,..,X_n$ be i.i.d continuous rvs. A record is said to occur at time $k$ if $X_k > X_i$ for all $i = 1,…,k-1$. Let $N$ denote the number of records. Find the variance […]

iid and correlated order statistics a comparison

Consider the two random variables $X_1$ and $X_2$ defined via i.i.d, non-negative random variables $Y_1$ and $Y_2$ as $ X_1=\min(Y_1,Y_2)\\$ $ X_2=\min(Y_1,Y_1)$ – (maximally correlated) The question is can we say $X_1 \leq X_2$ always? How to prove it?

Prove uniform distribution

For any random variable $X$, there exists a $U(0,1)$ random variable $U_X$ such that $X=F_X^{-1}(U_X)$ almost surely. Proof: In the case that $F_X$ is continuous, using $U_X=F_X(X)$ would suffice. In the general case, the statement is proven by using $U_X=F_X(X^-)+V(F_X(X)-F_X(X^-))$, where $V$ is a $U(0,1)$ random variable independent of $X$ and $F_X(x^-)$ denotes the left […]

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn’t find the mean and variance of the order statistics of a discrete uniform sample without replacement anywhere, so I figured I’d derive them here for future reference. Draw $k$ distinct numbers uniformly from […]

How to solve this confusing permutation problem related to arrangement of books?

Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the […]

Why the expected value of the error when doing regression by OLS is 0?

What I know to begin with is that the sum will be 0 if there is a y-intercept b0 , why is that? my book doesnt say and can’t figure it out. I also know that an importantant assumption for the OLS estimators to be BlUE is that x and the erros can’t be corralated […]

How to find distribution of order statistic

Let $X_i$ be iid random variables with common density $f$ and distribution $F$. Let $Y_k = X_{(k)}$ be the k-th order statistics (that is, $Y_1 = X_{(1)} = \min(X_i)$ etc.). Show that the joint density of the order statistics is given by $$f_{X_{(1)}, X_{(2)}, \dots, X_{(n)}}(y_1, \dots, y_n) = \begin{cases} n! \prod_{i=1}^n f(y_i) & \text{for […]

Expected maximum absolute value of $n$ iid standard Gaussians?

I have a problem where my errors are normally distributed and I want to know what the expected maximum error is if I repeat the process $n$ times. What is the smallest constant $C$ such that the following statement is true for all $n\geq 2$? Let $X_1, X_2, \cdots, X_n$ be independent standard Gaussian random […]

Conditional expectation to de maximum $E(X_1\mid X_{(n)})$

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1): Which is $E(X_1\mid X_{(n)})$ ? where $X_{(n)}=\max\{X_1,\ldots,X_n\}$

Covariance of order statistics (uniform case)

Let $X_1, \ldots, X_n$ be uniformly distributed on $[0,1]$ and $X_{(1)}, …, X_{(n)}$ the corresponding order statistic. I want to calculate $Cov(X_{(j)}, X_{(k)})$ for $j, k \in \{1, \ldots, n\}$. The problem is of course to calculate $\mathbb{E}[X_{(j)}X_{(k)}]$. The joint density of $X_{(j)}$ and $X_{(k)}$ is given by $$f_{X_{(j)}, X_{(k)}}=\binom{n}{k}\binom{k}{j-1}x^{j-1}(y-x)^{k-1-j}(1-y)^{n-k}$$ where $0\leq x\leq y\leq 1$. […]