The average of the items of $(0,10]$
$begingroup$
The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?
means
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
$endgroup$
|
show 1 more comment
$begingroup$
The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?
means
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
$endgroup$
1
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11
|
show 1 more comment
$begingroup$
The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?
means
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
$endgroup$
The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?
means
means
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
edited Dec 17 '18 at 21:02
tesryt ety
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
asked Dec 17 '18 at 20:50
tesryt etytesryt ety
83
83
1
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11
|
show 1 more comment
1
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11
1
1
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
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add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
$begingroup$
Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
$endgroup$
add a comment |
$begingroup$
Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
$endgroup$
add a comment |
$begingroup$
Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
$endgroup$
Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
answered Dec 17 '18 at 21:31
ChristophChristoph
12.3k1642
12.3k1642
add a comment |
add a comment |
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1
$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53
$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59
$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02
$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05
$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11