Linked Questions

0
votes
0answers
176 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
127 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)...
0
votes
0answers
45 views

Expected value of max(X,Y,Z) [duplicate]

Let X, Y, Z be independent normal variables, all with 0 mean and variance $\sigma_1,\sigma_2,\sigma_3$. What is the value of E(max(X,Y,Z))? I managed to solve it when there is just 2 of them, X and Y, ...
30
votes
2answers
15k 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 ...
29
votes
6answers
2k 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 ...
17
votes
2answers
12k views

Expected value for maximum of n normal random variable

Let $X_1...X_n\sim N(\mu,\sigma)$ be normal random variables. Find the expected value of $\max_i(X_i)$ and $\min_i(X_i)$. The sad truth is I don't have any good idea how to start and I'll be glad ...
22
votes
3answers
747 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 ...
2
votes
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
1k 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
4k 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 ...
8
votes
3answers
2k 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 $x_1$ ...
8
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 ...
4
votes
1answer
1k 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
2k 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) ...

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