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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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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?