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## Distribution of \$ text {arg max} _i vert X_i vert \$.

Leave $$X sim mathcal N (0.2)$$, $$X_2 = N + U$$, $$X_3 = N + V$$ where $$N, U, V sim mathcal N (0,1)$$. $$X_1, N, U, V$$ Independent. What is the distribution of $$text {arg max} _i vert X_i vert$$?

## r – How to correct the NA coercion error in the alpha function: "Error in FUN (X)[[i]], …): only defined in a data frame with all the numerical variables "

I'm trying to generate internal consistency by using the psych :: alpha function. I have executed it before, but now I am receiving a long error (attached) that ends with "Error in FUN (X)[[i]], …):
Only defined in a data frame with all numerical variables "

I have tried to subordinate the data set in a smaller data set with key variables and run alpha throughout the data frame: alpha (tData).

I also tried to coerce the variables into numeric: as.numeric

tData <- read.spss (file = "x", header = TRUE, to.data.frame = TRUE)

sleepTotal_alpha <- psych :: alpha (tData[ ,c("PresleepsametimeR", "Presleepwin20minsR", "PresleepaloneinbedR", "PreSleepParSibBed", "PreSleepRocking", "PreSleepParentinRoom", "PreSleepResistsBed", "PresleepsameamR", "PreSleepRestless", "PreSleepMovesBed", "PreSleepSnores", "PreSleepAwakensScreaming", "PreSleepNaps", "PresleepwakesonceR", "PreSleepWakesMore", "PreSleepWakesEarly", "PreSleepSeemsTired", "PreSleepfallsAsleep")], na.rm = TRUE)

print (sleepTotal_alpha)

The result is an error code instead of returning Cronbach's alpha. Halp!

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## probability: why is the event \$ {X_ {(j)} le x_i } \$ equivalent to the event \$ {Y_i ge j } \$?

From the statistical inference of Casella and Berger:

Leave $$X_1, points X_n$$ Be a random sample of a discrete distribution.
with $$f_X (x_i) = p_i$$, where $$x_1 lt x_2 lt dots$$ they are possible
values ​​of $$X$$ in ascending order. Leave $$X _ {(1)}, dots, X _ {(n)}$$
denotes the order statistics of the sample. Define $$Y_i$$ as the number of $$X_j$$ that are less than or equal to
$$x_i$$. Leave $$P_0 = 0, P_1 = p_1, dots, P_i = p_1 + p_2 + dots + p_i$$.

Yes $${X_j le x_i }$$ it's a "success" and $${X_j gt x_i }$$ it is a "failure", then $$Y_i$$ it is binomial with parameters $$(n, P_i)$$.

Then the event $${X _ {(j)} le x_i }$$ is equivalent to the event $${Y_i ge j }$$

Can anyone explain why these two are equivalent?

$${X_ {(j)} le x_i } = {s in text {sun} (X _ {(j)}): X _ {(j)} (s) le x_i }$$

$${Y_i ge j } = {s & # 39; in text {sun} (Y_i): Y_i (s & # 39;) ge j }$$

I have trouble understanding how these functions of random variables show this equivalence.

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## probability: find the p.d.f of \$ frac {x_1} { sum_ {i = 1} ^ n {(x_i)}} \$ where \$ x_i sim Exponetial ( theta) \$? and \$ x_i \$ are i.i.d?

How is the probability density function of the random variable derived? $$frac {x_1} { sum_ {i = 1} ^ n {(x_i)}}$$ where $$x_i sim Exponetial ( theta)$$? Y $$x_i$$ are i.i.d?

I can not think of the bivariate / multivariate transformation needed to do this.

I know how to expand the denominator, but I can not see the solution after expanding the denominator.

## reference request – Formula for the volume of \$ {x_i in [-N,N]: sum_ {i = 1} ^ n x_i = 0 } \$

I'm not an expert in convex geometry but if we define $$a_i sim mathcal {U} ([-N,N]$$ where $$[-N,N] subset mathbb {R}$$ Y $$S_n = sum_ {i = 1} ^ n a_i$$ I would like to know if for arbitrary $$N in mathbb {R} _ +$$:

1. The following limit is always true:

$$begin {equation} lim_ {n to infty} P (S_n = 0) = 0 tag {*} end {equation}$$

1. There is a simple formula for the volume:

$$begin {equation} Vol ( {x_i in [-N,N]: sum_ {i = 1} ^ n x_i = 0 }) end {equation}$$

So far I have managed to approach the discrete case by modeling it as a random walk in $$mathbb {Z}$$. Basically, I was able to prove that if we assume $$a_i sim mathcal {U} ([-N,N]$$ where $$[-N,N] subset mathbb {Z}$$ so:

$$begin {equation} lim_ {n to infty} P (S_n = 0) = 0 tag {1} end {equation}$$

$$begin {equation} lim_ {n a infty} P (| S_n | leq N) = 0 tag {2} end {equation}$$

$$begin {equation} lim_ {n a infty} P (| S_n |> N) = 1 tag {3} end {equation}$$

A description of my analysis is in my blog, Kepler Lounge. Having said that, I very much doubt that he is the first person to make this discovery.

I suspect that the result I am looking for is known to experts in convex analysis, but I do not know what references I should consult.

## functional analysis – \$ conv (A) = { sum_ {i = 1} ^ n lambda _i x_i: sum_ {i = 1} ^ n lambda_i = 1, lambda_i ge 0, x_i in A, i = 1, cdots, n } \$

Leave $$X$$ be a normed vector space and $$A$$ be a subset of $$X$$. $$conv (A)$$
It is called the intersection of all convex subsets of $$X$$ that contain
$$A$$

a) Show that conv (A) is a convex set

b) show that

$$conv (A) = { sum_ {i = 1} ^ n lambda _i x_i: sum_ {i = 1} ^ n lambda_i = 1, lambda_i ge 0, x_i in A, i = 1, cdots, n }$$

c) yes $$A$$ it is compact then it is $$conv (A)$$ compact?

d) Show that if $$A subseteq mathbb {R} ^ n$$ it is compact then $$conv (A)$$ is
compact

a) Take two elements $$a$$ Y $$b$$ at the intersection of all convex subsets of $$X$$ that contain $$A$$. Now take $$lambda a + (1- lambda) b$$. It is contained in each of the subsets of $$X$$ that contain $$A$$, therefore it is contained in the intersection. Q.E.D.

second)

$$leftarrow$$:

Suppose by induction that $$sum_ {i = 1} ^ n lambda_i x_i$$ It belongs to conv (A). We must prove that $$sum_ {i = 1} ^ {k + 1} lambda_i x_i$$, with $$sum_ {i = 1} ^ {k + 1} = 1$$ It also belongs.

$$sum_ {i = 1} ^ {k + 1} lambda_i x_i = sum_ {i = 1} ^ {k} lambda_i x_i + lambda_ {k + 1} x_ {k + 1} = sum_ {i = 1} ^ {k} lambda_i x_i + (1- sum_ {i = 1} ^ n lambda_i) x_ {k + 1}$$

Now choose $$delta$$ such that $$delta sum_ {i = 1} ^ {k} lambda_i = 1$$, so

$$frac { delta} { delta} left ( sum_ {i = 1} ^ {k} lambda_i x_i + (1- sum_ {i = 1} ^ n lambda_i) x_ {k + 1 } right = = left ( frac {1} { delta} sum_ {i = 1} ^ {k} ( delta lambda_i) x_i + frac {1} { delta} ( delta-1) x_ {k + 1} right) = \ frac {1} { delta} x + left (1- frac {1} { delta} right) x_ {k + 1}$$

which is a collection of elements of $$A$$ that adds up to $$1$$

$$rightarrow$$:

I can only say that $$x in conv (A)$$, so $$x = 1x + 0 cdot all$$ Therefore, it is a combination that adds to $$1$$ of elements of $$A$$?

by do), give me a clue: show that conv (A union B) is the image through a continuous function of the compact $${( alpha, beta; alpha, beta ge 0, alpha + beta = 1) } times A times B$$. I do not understand how to show this.

re) Some clue?