Is there an existing result about the density (in relation to the Lebesgue measure) $ mathbb {R} $) of the random variables:

$$ frac { sum_ {i = 1} ^ nX_i ^ 4} {( sum_ {i = 1} ^ nX_i ^ 2) ^ 2} $$

when $ X_i $ are iid $ N (0,1) $? Is it possible to get an analytic expression or at least a simple expression for the density function?

# Tag: iid

## Probability – limiting the distribution of the "scatter matrix" $ frac {1} {n} XX ^ T: = frac {1} {n} sum_ {i = 1} ^ nx_ix_i ^ T $ for iid $ x_1, ldots , x_n in mathbb R ^ p $

To let $ x_1, ldots, x_n $ iid be drawn from such a "nice" distribution $ mathbb R ^ p $ (but maybe very common!) and let $ X $ be that $ n $-by-$ p $ Matrix formed by vertical stacking $ x_i $& # 39; see Fig.

Question.What is the boundary distribution of the "scattering matrix" $ frac {1} {n} XX ^ T: = frac {1} {n} sum_ {i = 1} ^ nx_ix_i ^ T $ as $ n rightarrow infty $ ?

- If that $ x_i $So they come from a centered multivariate Gaussian $ XX ^ T $ has a Wishart distribution.

## Probability – Need help to correct / clarify my considerations on iid RVs after learning some statistics for the first year

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## Linux – Configure the IPv6 address at the interface to the static IID

I'm looking for a tool similar to rdisc6 that configures the v6 address (s) on an interface using a static IID upon receiving the RA. This is a server that must be located at a known address within a ULA. (No, I can not use mDNS and SLAAC because name-bound certificates exist and mDNS may not work unless this interface is configured.)

If I have to, I will expand rdisc6, but I hope to not replicate something that someone has already done.

This is done under Linux (armv7) in an LXC container.

## Probability – How will these expectation values โโ(inner product values) be obtained between iid functions?

Consider the following two polynomials:

$ f_1 = x_1 ^ 2-1 $ and $ f_2 = x_1 ^ 2x_2 ^ 2 + x_1 ^ 2-x_2 ^ 2 + 1 $,

Both are Hermite polynomials $ x_1 $ and $ x_2 $ are independent normal variables with a mean of 0 and a variance of 1.

How can we achieve these expectations (inner product over the Hilbert space) if we know that? $

$

$

$

Is there a way to broaden that expectation formula and be able to calculate the expectations without having to integrate the inner product?

## Probability – Joint distribution of two weighted sums of IID random variables

To let $ X_1, X_2, points $ independently distributed random variables in $ {- 1, +1 } $ and let it go $ a_1, b_1, a_2, b_2, ldots in mathbb {R} $ fixed, limited and non-zero mean. To let $ Y_n = a_1X_1 + cdots + a_nX_n $ and $ Z_n = b_1X_1 + cdots + b_nX_n $,

I am interested in understanding the common dissemination of $ Y_n $ and $ Z_n $ as $ n in infty $ and more exactly, if you give an upper bound for the probability

$$ mathbb {P}[|Y_n| leq x land |Z_n| leq y], $$

that is uniform in $ x, y in mathbb {R} $ and $ n in mathbb N $, A quick calculation seems to show the distribution and a limitation of the form

$$ mathbb {P}[|Y_n| leq x land |Z_n| leq y]= O ! Left ( frac {(1 + x) (1 + y)} {n} right) $$

should be achievable if the sequences $ a_1, a_2, dots $ and $ b_1, b_2, dots $ are sufficiently "independent" of each other.

I could not find anything in the literature, but I guess these kinds of problems should have been thoroughly investigated, so I'm looking for clues. Many Thanks

## Probability – Determine the CDF of a sum of contiguous random I.I.D variables

Be X$ _1 $ and X$ _2 $ identically independent distributions (i.i.d) using random variables

P (X$ _i $ $ le $ x) = 1-x$ ^ {- 1/3} $

x $ ge $ 1 and i = 1.2

Find P (X$ _1 $ + X$ _2 $ $ le $ x)

I tried to find the convolution of f$ _ {X_1} $ and f$ _ {X_2} $ (the density functions for X$ _1 $ and X$ _2 $) and integrate to get the CDF of X.$ _1 $ + X$ _2 $ That's what I interpreted the question. When I integrated it, it got very messy and I could not finish the integration. Am I doing something wrong?

## Probability – When the arrival times of the customers i.i.d. exponential distribution, is it necessary that the number of customers is a Poisson process?

For example, suppose customers arrive at a time interval $ U_i $ i.i.d. $ Exp ( lambda) $, that's why,

$$ F (U_i le t) = 1-e ^ {- lambda t} $$

The arrival time of the customer $ i $ is

$$ T_i = sum ^ i_ {j = 1} {U_j} $$

The number of customers who arrived on time $ t $ That's why

$$ N (t) = sum ^ infty_ {i = 1} {1 _ { {T_i le t }}}}

$ N (t) $ is a counting process.

$ N (t) $ is called the inhomogeneous Poisson counting process if it has the four properties.

This condition seems to indicate that i. The exponentially distributed arrival time does not guarantee that it is a Poisson process unless it meets the four characteristics.

1) Does this mean a probability distribution of $ N (t) $ is not clear or without assumptions like the four characteristics comprehensible?

2) If 1) is true, there is another possible formulation process besides Poisson $ N (t) $,

## st.statistics – concentration of $ X ^ T eta ^ TX in mathbb R ^ d $ for i.i.d $ (x_i, eta_i) $ and sub-gaussian $ eta_i $

Accept $ (x_1, eta_1), ldots, (x_n, eta_n) $ are $ n $ i.i.d shows in $ mathbb R ^ {d + 1} $ so that $ eta_1, ldots, eta_n $ are $ sigma $-subgaussian. To let $ X in mathbb R ^ {n times d} $ let be the vertical stacking of $ x_i $and $ eta in mathbb R ^ n $ let be the vertical stacking of $ eta_i $& # 39; s

Are there any concentration inequalities that can be made to bind the matrix? $ X ^ T eta eta ^ TX in mathbb R ^ {d times d} $ ?

Naively, I would guess that $ X ^ T eta eta ^ TX preceq sigma ^ 2X ^ TX + text {"little thing"} $, with high probability.

## pr.probability – maximum of the sums of iid $ X_i $ s, where $ X_i $ is the difference between two exponential values โโof r.v.

given $ X_i = A_i – B_i $ from where $ A_i sim text {Exp} ( alpha) $ and $ B_i sim text {Exp} ( lambda) $, Define $ S_k = sum_ {i = 1} ^ k X_i $ With $ S_0 = 0 $, and

$$ M_n = max_ {1 leq k leq n} S_k. $$

Can you calculate the amount? $ mathbb {P} (M_n leq x) $ expressly? I tried it and the result is down, but it is not clear …

In Feller's Introduction to Probability Theory and its application, he has repeatedly remarked that this type of two-exponential divergence distribution is a rare but important case in which almost all erroneous calculations can be made explicit. (V1.8 example (b) page 193; XII.2 example (b) page 395; XII.3 example (b) page 401) Unfortunately, I could not find any detailed calculations in the book.

A second reference that I have looked at is the paper "*On the distribution of the maximum of sums of independent and equally distributed random variables*"by Lajos Takacs (Adv. Appl. Prob 1970) Takacs mentioned that we can calculate in some special cases $ mathbb {P} (M_n leq x) $ light. After his example on page 346 (where he only suspected) $ X_i = A_i – B_i $ from where $ B_i $ is exponential and $ A_i $ is not negative) that I could count on $ A_i sim text {Exp} ( alpha) $, $ B_i sim text {Exp} ( lambda) $, we have

$$ U (s, p) = sum_ {n = 0} ^ infty mathbb {E} left[e^{-sM_n}right]p ^ n = frac { lambda – frac {s lambda} { gamma (p)}} { lambda – s – frac { lambda alpha p} { alpha + s}} $$

from where $ gamma (p) = frac { lambda – alpha + sqrt {( alpha + lambda) ^ 2 – 4 alpha lambda p}} {2} $a zero of the denominator above. Is there a way to simplify this to get an explicit formula for $ mathbb {P} (M_n leq x) $?