Mixing
Standard probability space and random variables.
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1
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Problem :
Let $Omega:=[0,1]$
How to define two independent random variables, which describe coin toss on the above omega?
My idea:
$Omega=[0,1]$
$Sigma = mathbb{B}([0,1])$
Consider the Lebesgue measure
1-reverse coin
2-obverse coin
$X(w)= begin{cases} 0 &text{when } w in [0,1/2) \ 1 &text{when } win [1/2,1) end{cases} $
$Y(w)= begin{cases} 1 &text{when } w in [0,1/4) cup [1/2,3/4) \ 0 &text{when } win [1/4,1/2)cup [3/4,1) end{cases} $
It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$
And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$
What do you think about this?
probability-theory random-variables
asked yesterday
PabloZ392
1356
add a comment |
up vote
1
down vote
favorite
Problem :
Let $Omega:=[0,1]$
How to define two independent random variables, which describe coin toss on the above omega?
My idea:
$Omega=[0,1]$
$Sigma = mathbb{B}([0,1])$
Consider the Lebesgue measure
1-reverse coin
2-obverse coin
$X(w)= begin{cases} 0 &text{when } w in [0,1/2) \ 1 &text{when } win [1/2,1) end{cases} $
$Y(w)= begin{cases} 1 &text{when } w in [0,1/4) cup [1/2,3/4) \ 0 &text{when } win [1/4,1/2)cup [3/4,1) end{cases} $
It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$
And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$
What do you think about this?
probability-theory random-variables
asked yesterday
PabloZ392
1356
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Problem :
Let $Omega:=[0,1]$
How to define two independent random variables, which describe coin toss on the above omega?
My idea:
$Omega=[0,1]$
$Sigma = mathbb{B}([0,1])$
Consider the Lebesgue measure
1-reverse coin
2-obverse coin
$X(w)= begin{cases} 0 &text{when } w in [0,1/2) \ 1 &text{when } win [1/2,1) end{cases} $
$Y(w)= begin{cases} 1 &text{when } w in [0,1/4) cup [1/2,3/4) \ 0 &text{when } win [1/4,1/2)cup [3/4,1) end{cases} $
It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$
And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$
What do you think about this?
probability-theory random-variables
asked yesterday
PabloZ392
1356
Problem :
Let $Omega:=[0,1]$
How to define two independent random variables, which describe coin toss on the above omega?
My idea:
$Omega=[0,1]$
$Sigma = mathbb{B}([0,1])$
Consider the Lebesgue measure
1-reverse coin
2-obverse coin
$X(w)= begin{cases} 0 &text{when } w in [0,1/2) \ 1 &text{when } win [1/2,1) end{cases} $
$Y(w)= begin{cases} 1 &text{when } w in [0,1/4) cup [1/2,3/4) \ 0 &text{when } win [1/4,1/2)cup [3/4,1) end{cases} $
It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$
And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$
What do you think about this?
probability-theory random-variables
probability-theory random-variables
asked yesterday
PabloZ392
1356
asked yesterday
PabloZ392
1356
edited yesterday
asked yesterday
PabloZ392
1356
asked yesterday
PabloZ392
1356
asked yesterday
PabloZ392
1356
1356
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1 Answer
1
active
oldest
votes
up vote
1
down vote
Yes, your idea works just fine.
answered yesterday
Nate Eldredge
61.4k679166
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Yes, your idea works just fine.
answered yesterday
Nate Eldredge
61.4k679166
add a comment |
up vote
1
down vote
Yes, your idea works just fine.
answered yesterday
Nate Eldredge
61.4k679166
add a comment |
up vote
1
down vote
up vote
1
down vote
Yes, your idea works just fine.
answered yesterday
Nate Eldredge
61.4k679166
Yes, your idea works just fine.
answered yesterday
Nate Eldredge
61.4k679166
answered yesterday
Nate Eldredge
61.4k679166
answered yesterday
Nate Eldredge
61.4k679166
answered yesterday
Nate Eldredge
61.4k679166
61.4k679166
add a comment |
add a comment |
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