Expectation of 1/x, x uniform from 0 to 1
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It seems to me that this expectation is infinite.
My argument takes a smaller function that is discreet. f(x) = 1 if x>.5, f(x)=2 if x>.25 and so on.
We can use linearity of expectation to calculate E(f(x))=1+1/2*1+1/4*2+1/8*4...
(where each term is the probability of passing a threshold and the increase for that threshold).
Is this correct? The reason I suspect if might not be: if I take a 0-1 uniform random matrix and do elimination, the second pivot seems to have distribution x4-(x3/x1)*x2, where x1,x2,x3 and x4 are uniform 0-1 and, therefore, infinite expected value (or at least infinite modulus)
linear-algebra expected-value
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
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add a comment |
$begingroup$
It seems to me that this expectation is infinite.
My argument takes a smaller function that is discreet. f(x) = 1 if x>.5, f(x)=2 if x>.25 and so on.
We can use linearity of expectation to calculate E(f(x))=1+1/2*1+1/4*2+1/8*4...
(where each term is the probability of passing a threshold and the increase for that threshold).
Is this correct? The reason I suspect if might not be: if I take a 0-1 uniform random matrix and do elimination, the second pivot seems to have distribution x4-(x3/x1)*x2, where x1,x2,x3 and x4 are uniform 0-1 and, therefore, infinite expected value (or at least infinite modulus)
linear-algebra expected-value
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
$endgroup$
add a comment |
$begingroup$
It seems to me that this expectation is infinite.
My argument takes a smaller function that is discreet. f(x) = 1 if x>.5, f(x)=2 if x>.25 and so on.
We can use linearity of expectation to calculate E(f(x))=1+1/2*1+1/4*2+1/8*4...
(where each term is the probability of passing a threshold and the increase for that threshold).
Is this correct? The reason I suspect if might not be: if I take a 0-1 uniform random matrix and do elimination, the second pivot seems to have distribution x4-(x3/x1)*x2, where x1,x2,x3 and x4 are uniform 0-1 and, therefore, infinite expected value (or at least infinite modulus)
linear-algebra expected-value
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
$endgroup$
It seems to me that this expectation is infinite.
My argument takes a smaller function that is discreet. f(x) = 1 if x>.5, f(x)=2 if x>.25 and so on.
We can use linearity of expectation to calculate E(f(x))=1+1/2*1+1/4*2+1/8*4...
(where each term is the probability of passing a threshold and the increase for that threshold).
Is this correct? The reason I suspect if might not be: if I take a 0-1 uniform random matrix and do elimination, the second pivot seems to have distribution x4-(x3/x1)*x2, where x1,x2,x3 and x4 are uniform 0-1 and, therefore, infinite expected value (or at least infinite modulus)
linear-algebra expected-value
linear-algebra expected-value
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
edited Dec 8 '18 at 16:11
josinalvo
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
asked Dec 8 '18 at 15:43
josinalvojosinalvo
1,153710
1,153710
add a comment |
add a comment |
1 Answer
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$$int_0^1 frac1x , dx = lim_{t to 0^+} int_t^1 frac1x , dx = lim_{t to 0^+}-log(t)= infty$$
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
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1 Answer
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1 Answer
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$begingroup$
$$int_0^1 frac1x , dx = lim_{t to 0^+} int_t^1 frac1x , dx = lim_{t to 0^+}-log(t)= infty$$
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
add a comment |
$begingroup$
$$int_0^1 frac1x , dx = lim_{t to 0^+} int_t^1 frac1x , dx = lim_{t to 0^+}-log(t)= infty$$
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
add a comment |
$begingroup$
$$int_0^1 frac1x , dx = lim_{t to 0^+} int_t^1 frac1x , dx = lim_{t to 0^+}-log(t)= infty$$
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$$int_0^1 frac1x , dx = lim_{t to 0^+} int_t^1 frac1x , dx = lim_{t to 0^+}-log(t)= infty$$
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 8 '18 at 15:47
Siong Thye GohSiong Thye Goh
101k1466118
101k1466118
add a comment |
add a comment |
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