How can I calculate $E[max ( X,1-X)]$ if $Xsim U(0,1)$? [closed]
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Given $ X sim U(0,1) $ and $Y=max X,1-X$, how can I calculate $E[Y]$ ?
probability probability-distributions uniform-distribution expected-value
closed as off-topic by Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist Sep 9 at 14:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist
add a comment |Â
up vote
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Given $ X sim U(0,1) $ and $Y=max X,1-X$, how can I calculate $E[Y]$ ?
probability probability-distributions uniform-distribution expected-value
closed as off-topic by Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist Sep 9 at 14:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist
What did you try?
â StubbornAtom
Sep 9 at 10:40
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
2
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given $ X sim U(0,1) $ and $Y=max X,1-X$, how can I calculate $E[Y]$ ?
probability probability-distributions uniform-distribution expected-value
Given $ X sim U(0,1) $ and $Y=max X,1-X$, how can I calculate $E[Y]$ ?
probability probability-distributions uniform-distribution expected-value
probability probability-distributions uniform-distribution expected-value
edited Sep 9 at 10:40
StubbornAtom
4,11711136
4,11711136
asked Sep 9 at 10:31
Software_t
1184
1184
closed as off-topic by Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist Sep 9 at 14:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist
closed as off-topic by Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist Sep 9 at 14:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Did, amWhy, José Carlos Santos, StubbornAtom, Theoretical Economist
What did you try?
â StubbornAtom
Sep 9 at 10:40
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
2
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02
add a comment |Â
What did you try?
â StubbornAtom
Sep 9 at 10:40
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
2
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02
What did you try?
â StubbornAtom
Sep 9 at 10:40
What did you try?
â StubbornAtom
Sep 9 at 10:40
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
2
2
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02
add a comment |Â
4 Answers
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4
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Well, observe that $Y=X$ if $X>frac12$ and $Y=1-X$ otherwise.
Therefore $$mathbbE[Y] = int_0^frac12 (1-x) dx + int_frac12^1 x dx = frac12 - frac18 + frac12 - frac18 = frac34 $$
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
add a comment |Â
up vote
2
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$Y$ is uniform in $[0.5,1]$. So, $mathbbE[Y]$ is $0.75$
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
add a comment |Â
up vote
1
down vote
In general if $X$ is a random variable having a PDF $f_X$ then for Borel-measurable function $g:mathbb Rtomathbb R$ we have:$$mathbb Eg(X)=int g(x)f_X(x);dx$$if the expectation exists.
That can be applied here on function $g$ prescribed by $xmapstomax(x,1-x)$.
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
add a comment |Â
up vote
1
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$$mathbb E[Y] = mathbb Eleft[Y mid X>tfrac12right]mathbb Pleft(X>tfrac12right) + mathbb Eleft[Y mid Xle tfrac12right]mathbb Pleft(Xletfrac12right)$$ $$= tfrac12 mathbb Eleft[X mid X>tfrac12right] + tfrac12 mathbb Eleft[1-X mid 1-Xge tfrac12right] $$ $$= frac12 times frac34 + frac12 times frac34 = frac34 $$
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Well, observe that $Y=X$ if $X>frac12$ and $Y=1-X$ otherwise.
Therefore $$mathbbE[Y] = int_0^frac12 (1-x) dx + int_frac12^1 x dx = frac12 - frac18 + frac12 - frac18 = frac34 $$
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
add a comment |Â
up vote
4
down vote
Well, observe that $Y=X$ if $X>frac12$ and $Y=1-X$ otherwise.
Therefore $$mathbbE[Y] = int_0^frac12 (1-x) dx + int_frac12^1 x dx = frac12 - frac18 + frac12 - frac18 = frac34 $$
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Well, observe that $Y=X$ if $X>frac12$ and $Y=1-X$ otherwise.
Therefore $$mathbbE[Y] = int_0^frac12 (1-x) dx + int_frac12^1 x dx = frac12 - frac18 + frac12 - frac18 = frac34 $$
Well, observe that $Y=X$ if $X>frac12$ and $Y=1-X$ otherwise.
Therefore $$mathbbE[Y] = int_0^frac12 (1-x) dx + int_frac12^1 x dx = frac12 - frac18 + frac12 - frac18 = frac34 $$
answered Sep 9 at 10:36
LucaMac
1,694116
1,694116
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
add a comment |Â
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
+1 This is in my view the best answer to the need of OP. Especially because it is direct and only requires basic knowledge of the OP.
â drhab
Sep 9 at 11:23
add a comment |Â
up vote
2
down vote
$Y$ is uniform in $[0.5,1]$. So, $mathbbE[Y]$ is $0.75$
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
add a comment |Â
up vote
2
down vote
$Y$ is uniform in $[0.5,1]$. So, $mathbbE[Y]$ is $0.75$
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
add a comment |Â
up vote
2
down vote
up vote
2
down vote
$Y$ is uniform in $[0.5,1]$. So, $mathbbE[Y]$ is $0.75$
$Y$ is uniform in $[0.5,1]$. So, $mathbbE[Y]$ is $0.75$
edited Sep 9 at 10:38
LucaMac
1,694116
1,694116
answered Sep 9 at 10:33
gunes
1943
1943
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
add a comment |Â
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
Can you explain that?
â Software_t
Sep 9 at 10:33
Can you explain that?
â Software_t
Sep 9 at 10:33
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
if you have X = 0.1, or 0.9, both contributes to Y = 0.9. So, for each Y, you have two X contributors. This leads to uniform distribution among Y.
â gunes
Sep 9 at 10:35
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
For each $yinleft(frac12,1right)$, one can find the distribution function of $Y$ as beginalign P(Yle y)&=1-P(Y>y) \&=1-Pleft[X>ycup1-X>yright] \&=1-P(X>y)-P(X<1-y) \&=cdots endalign , which would prove that $Ysim Uleft(frac12,1right)$.
â StubbornAtom
Sep 9 at 10:47
add a comment |Â
up vote
1
down vote
In general if $X$ is a random variable having a PDF $f_X$ then for Borel-measurable function $g:mathbb Rtomathbb R$ we have:$$mathbb Eg(X)=int g(x)f_X(x);dx$$if the expectation exists.
That can be applied here on function $g$ prescribed by $xmapstomax(x,1-x)$.
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
add a comment |Â
up vote
1
down vote
In general if $X$ is a random variable having a PDF $f_X$ then for Borel-measurable function $g:mathbb Rtomathbb R$ we have:$$mathbb Eg(X)=int g(x)f_X(x);dx$$if the expectation exists.
That can be applied here on function $g$ prescribed by $xmapstomax(x,1-x)$.
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
add a comment |Â
up vote
1
down vote
up vote
1
down vote
In general if $X$ is a random variable having a PDF $f_X$ then for Borel-measurable function $g:mathbb Rtomathbb R$ we have:$$mathbb Eg(X)=int g(x)f_X(x);dx$$if the expectation exists.
That can be applied here on function $g$ prescribed by $xmapstomax(x,1-x)$.
In general if $X$ is a random variable having a PDF $f_X$ then for Borel-measurable function $g:mathbb Rtomathbb R$ we have:$$mathbb Eg(X)=int g(x)f_X(x);dx$$if the expectation exists.
That can be applied here on function $g$ prescribed by $xmapstomax(x,1-x)$.
edited Sep 9 at 10:36
answered Sep 9 at 10:34
drhab
89.3k541123
89.3k541123
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
add a comment |Â
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I know that, but how you will define the function $g(x)$ in this case?
â Software_t
Sep 9 at 10:36
I have added that.
â drhab
Sep 9 at 10:37
I have added that.
â drhab
Sep 9 at 10:37
add a comment |Â
up vote
1
down vote
$$mathbb E[Y] = mathbb Eleft[Y mid X>tfrac12right]mathbb Pleft(X>tfrac12right) + mathbb Eleft[Y mid Xle tfrac12right]mathbb Pleft(Xletfrac12right)$$ $$= tfrac12 mathbb Eleft[X mid X>tfrac12right] + tfrac12 mathbb Eleft[1-X mid 1-Xge tfrac12right] $$ $$= frac12 times frac34 + frac12 times frac34 = frac34 $$
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
add a comment |Â
up vote
1
down vote
$$mathbb E[Y] = mathbb Eleft[Y mid X>tfrac12right]mathbb Pleft(X>tfrac12right) + mathbb Eleft[Y mid Xle tfrac12right]mathbb Pleft(Xletfrac12right)$$ $$= tfrac12 mathbb Eleft[X mid X>tfrac12right] + tfrac12 mathbb Eleft[1-X mid 1-Xge tfrac12right] $$ $$= frac12 times frac34 + frac12 times frac34 = frac34 $$
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
add a comment |Â
up vote
1
down vote
up vote
1
down vote
$$mathbb E[Y] = mathbb Eleft[Y mid X>tfrac12right]mathbb Pleft(X>tfrac12right) + mathbb Eleft[Y mid Xle tfrac12right]mathbb Pleft(Xletfrac12right)$$ $$= tfrac12 mathbb Eleft[X mid X>tfrac12right] + tfrac12 mathbb Eleft[1-X mid 1-Xge tfrac12right] $$ $$= frac12 times frac34 + frac12 times frac34 = frac34 $$
$$mathbb E[Y] = mathbb Eleft[Y mid X>tfrac12right]mathbb Pleft(X>tfrac12right) + mathbb Eleft[Y mid Xle tfrac12right]mathbb Pleft(Xletfrac12right)$$ $$= tfrac12 mathbb Eleft[X mid X>tfrac12right] + tfrac12 mathbb Eleft[1-X mid 1-Xge tfrac12right] $$ $$= frac12 times frac34 + frac12 times frac34 = frac34 $$
answered Sep 9 at 11:01
Henry
94.4k473150
94.4k473150
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
add a comment |Â
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
Always a bit odd to see solutions based on the premise that conditional probabilities/expectations would be simpler to grasp than absolute probabilities/expectations.
â Did
Sep 9 at 14:08
add a comment |Â
What did you try?
â StubbornAtom
Sep 9 at 10:40
Actually the question is just a special case of the more general version: How can I calculate $mathbb Eg(X)$ if $X$ is a random variable that has a PDF. IMHO the knowledge of how to do that in general has more value than the knowledge how to do that in some special case.
â drhab
Sep 9 at 10:56
@drhab Yes, this is just about evaluating an integral.
â StubbornAtom
Sep 9 at 11:00
2
@StubbornAtom Then why not directly asking for evaluation of $int_0^1max(x,1-x) dx$?
â drhab
Sep 9 at 11:02