Linked Questions

0
votes
0answers
88 views

expectation of maximum of iid random variables from normal distribution. [duplicate]

1) How to find expectation of max of random variables , i.e : $\mathbb{E}[max(x_1,x_2,\dots,x_n)]$ where $x$ are IID random variables from $\mathcal{N}(\mu,\sigma^2)$. I know that CDF is $F(x)^n$ ...
1
vote
0answers
46 views

$\max(X_1, X_2, … X_n)$ mean value [duplicate]

Suppose $X_1, X_2 ...X_n$ are non-correlated and Gaussian random variables, all with mean value $\mu=0$ and variance=$\sigma$. Is there an expression for the distribution of $Z=\max(X_1, X_2, ... X_n)...
24
votes
2answers
12k views

Does convergence in distribution implies convergence of expectation?

If we have a sequence of random variables $X_1,X_2,\ldots,X_n$ converges in distribution to $X$, i.e. $X_n \rightarrow_d X$, then is $$ \lim_{n \to \infty} E(X_n) = E(X) $$ correct? I know that ...
28
votes
6answers
1k views

Does exceptionalism persist as sample size gets large?

Which of the following is more surprising? In a group of 100 people, the tallest person is one inch taller than the second tallest person. In a group of one billion people, the tallest person is one ...
22
votes
3answers
676 views

How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how ...
1
vote
2answers
3k views

Bounds for the maximum of binomial random variables

Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. How can one derive lower and upper bounds for the expected value of the maximum of $n$ such random variables? I am ...
7
votes
2answers
687 views

A reference for a Gaussian inequality ($\mathbb{E} \max_i X_i$)

I am looking for a reference to cite, for the following "folklore" asymptotic behaviour of the maximum of $n$ independent Gaussian real-valued random variables $X_1,\dots, X_n\sim \mathcal{N}(0,\sigma)...
2
votes
3answers
3k views

What is the maximum expected value in a selection?

Here's a hypothetical problem: assume that mean diameter of a tennis ball is 6.7 cm. Assume that the diameter is normally distributed with a standard deviation of 0.1 cm (I may have picked up a weird ...
7
votes
3answers
1k views

Variance of a max function

Say $x_1$ and $x_2$ are normal random variables with known means and standard deviations and $C$ is a constant. If $y = \max(x_1,x_2,C)$, what is $\mathrm{Var}(y)$? Well, I forgot to tell that $x_1$ ...
5
votes
1answer
1k views

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 ...
3
votes
1answer
774 views

Distribution of largest sample from normal distribution.

Given $n$ independent random variables $X_i$ with normal distribution, mean $\mu$, variance $\sigma^2$, what is the distribution of $\max\limits_{i=1}^n(X_i)$ ? In particular I am interested in ...
5
votes
1answer
1k views

Range of i.i.d. normal random variables

Let $X_1, \dotsc, X_n$ be i.i.d. standard normal random variables. Define the range $R \in \mathbb{R}_{\geq 0}$ as $R = \max \{X_1, \dotsc, X_n\} - \min \{X_1, \dotsc, X_n \}$. I am looking for a ...
1
vote
1answer
1k views

PDF for minimum and maximum of n independent Gaussian random variables

Stimulated by the problem Let $Z\sim N(0,1)$ be a random variable, then $E[\max\{Z,0\}]$ is? I came up with this problem: Let $x_i, i=1..n$ be $n$ independent random variables $\sim N(0,1)$. 1) ...
4
votes
2answers
201 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with a normal distribution with the mean 0 and variance $m$) to its diagonal. A is a square ...
2
votes
0answers
226 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: $$\tag{1}\...

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