All Questions
Tagged with extreme-value random-variable
23 questions
3
votes
1
answer
85
views
Maximum of two independent gamma variables
Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$.
Are there known results for the distribution of $X$? Actually I only need to know $\...
- 1,191
0
votes
0
answers
25
views
Linearity of and pointwise equality in expectation of min() function
Consider the expressions $f = c + s*E[min(a/s, X)]$ and $g = E[min(c + a, c+sX)]$ where
c >= 0
0 < s <= 1
a >= 0
X ~ Poisson($\lambda$/s)
I'd like to think that $f = g$, reasoning as ...
5
votes
2
answers
346
views
What is the median of the minimum or maximum of multiple samples?
Suppose I have a variable with a known distribution, and suppose I sample that variable k times and record the minimum. If I repeat this many times, will the median of the minimum converge to a ...
- 55
2
votes
1
answer
52
views
How to assign reasonable scale parameters to randomly generated intercepts for the Weibull distribution?
This is a follow-on to post Correctly simulating an extreme value distribution for survival analysis?, as I work towards adaptation of that code to the Weibull distribution. In the below code I ...
- 827
8
votes
4
answers
1k
views
Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193
0
votes
0
answers
76
views
Normalization of $M_{n} = \max(U_{1}, ... , U_{n})$
Let $M_{n} = \max(U_{1}... , U_{n})$ be the maximum of a sample size $n$ from $U(0 , 1)$ distribution.
In my statistics textbook it says that $M_{n}$ normalized is equal to $n(1 - M_{n})$ but I'm not ...
- 107
4
votes
1
answer
1k
views
Distribution of the second minimum of a set of random variables
Given $n$ i.i.d. random variables $X_1,...X_n$, what is the distribution of the second smallest value ?
I know from this question that CDF of the minimum value is $1 - (1-F(x))^n$ where $F(x)$ is the ...
- 155
3
votes
3
answers
1k
views
Expectation of the minimum of a continuous random variable $X$ and a discrete random variable $Y$
Let $X\sim Exp(1)$ and independently let $Y$ have the pmf $P(Y=k)= p$, $P(Y = \infty) = 1-p$, where $k < \infty$. I'd like to calculate $\mathbb{E}(Z)$, where $Z = \min(X,Y)$.
Usually, we tackle ...
- 309
1
vote
0
answers
88
views
How to compute largest values of random variables? [closed]
Suppose we have two discrete random variables and we want perform maximum operation to obtain the max PDF.
We know that max of two independent random variables is:
if Z = max(X,Y)
...
- 11
4
votes
1
answer
6k
views
Cdf of minimum of two iid random variables
I am struggling with the following sentence:
Using the fact that the cumulative distribution of the
minimum of two i.i.d. random variables can be expressed as $1 - (1 - F(x))^2$....
Can anyone ...
- 694
1
vote
0
answers
88
views
Maximise the probability of a linear combination of random variables
I have a data set representing a random vector $\mathbf{X}=(X_1,\ldots, X_p)'$.
Define $Z=\alpha' X $, where $\alpha \in \mathbb{R}^p$ and $\alpha'\mathbf{1}_p =1$.
I would like to find the $\alpha$ ...
- 163
3
votes
1
answer
353
views
How does the maximum distance between adjacent values vary for increasing $n$
That is, when is the $\underset{n \to \infty}{\lim} \max (X_i-X_{i-1})\rightarrow 0$, where $1<i\leq n$, and $X_i\geq X_{i-1}$ and when is the limit $\neq 0$? The question supposes that the ...
- 13.3k
3
votes
1
answer
2k
views
How do I compare the the sampling distribution of the minimum of a distribution by sample sizes
I saw this question (link) but when I read it, I see that it has a fixed "N" so I thought it was asking about for a finite sample size. When I read the answer that it was suggested to be a duplicate ...
- 9,853
1
vote
2
answers
370
views
Why is $P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$?
Why is $$P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$$
and similarly
$$P(\max\{X_1,...,X_n\}≤y)=P(X_1≤y, ..., X_n≤y)$$
I.e. why are $\min$ and $\max$ equivalent to AND (since $P(X_1≥y, X_2≥y)$ ...
- 4,129
0
votes
1
answer
2k
views
Throwing two four-sided dice; min/max problems
Suppose two four-sided fair dice. Let $X = min(X_{1}, X_{2})$ and $Y = max(X_{1}, X_{2})$ where $X_{1}$ is the result of the first dice and $X_{2}$ the result of the second dice. What is the PMF of ...
- 1
2
votes
1
answer
7k
views
Proof for the p.d.f of minimum and maximum of a sample
The following is a question from a past paper for one of my university statistical inference modules, and I know how to use the formula for each the max/min, but
Assume that the sample $X_1, X_2, ....
- 23
1
vote
1
answer
70
views
Exclude Some samples for calculating CDF
I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and $p&...
- 301
5
votes
1
answer
400
views
Properties of the minimum of several random variables
I've come across an interesting problem in my research that I don't quite know the answer to. Suppose I have a bunch of random variables:
$$ X_1, X_2, X_3, ... X_N $$
They are not identical but they ...
- 311
1
vote
1
answer
2k
views
Probability density function of the sample maximum of a random variable
According to my book, for a random sample $(X_1, \ldots, X_n)$ from a continuous distribution with p.d.f. $f(x)$ and c.d.f. $F(x)$, the p.d.f. of the maximum of the sample is $g(z)=nf(z)[F(z)]^{n-1}$, ...
- 13
2
votes
0
answers
1k
views
maximum gap between order statistics of normally distributed random variables [closed]
I am currently working on a not-that-easy problem involving order statistics. As I am unsure as to how I could solve it, I thought it might already possess a solution. So here I am, my questions is: ...
- 21
2
votes
2
answers
25k
views
Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables
Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval $(-1,1)$.
Determine the density for $U=\min(X,Y)$ and for $W=\max(X,Y)$
- 23
14
votes
4
answers
1k
views
Unbiased estimator for the smaller of two random variables
Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$
I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$?
The simple estimator ...
- 141
83
votes
3
answers
105k
views
How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
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