Mixing
How to compute probability density fuction of a continuous function and then expected value of it?
$begingroup$
I have this discontinuous function:
$$f(r_i) =
begin{cases}
1, & text{if}~ |r_i| < 1.5 \
0, & text{if}~ |r_i| >1.5
end{cases}$$
where $r_i = z_i - a^2_ihat{x}~$, with $z_i$ being the $i$th column of matrix $z = Hx + e$, which error component $esim mathcal{N}(0, mathcal{I})$ has a covariance $mathcal{I}$ being the identity matrix, and $a_i$ is the $i$th column vector of a given matrix $H$.
I want to compute probability density function of it and then expectation value. How can I do that (calculating pdf function of these functions)?
What about this slightly different function below with the same $r_i$ ?
$$f(r_i) =
begin{cases}
r_i^2, & text{if}~ |r_i| < 1.5 \
(1.5)^2 sgn^2(r_i), & text{if}~ |r_i| > 1.5
end{cases}$$
probability-distributions expected-value
asked Jan 26 at 13:11
neda neda
11
$endgroup$
add a comment |
$begingroup$
I have this discontinuous function:
$$f(r_i) =
begin{cases}
1, & text{if}~ |r_i| < 1.5 \
0, & text{if}~ |r_i| >1.5
end{cases}$$
where $r_i = z_i - a^2_ihat{x}~$, with $z_i$ being the $i$th column of matrix $z = Hx + e$, which error component $esim mathcal{N}(0, mathcal{I})$ has a covariance $mathcal{I}$ being the identity matrix, and $a_i$ is the $i$th column vector of a given matrix $H$.
I want to compute probability density function of it and then expectation value. How can I do that (calculating pdf function of these functions)?
What about this slightly different function below with the same $r_i$ ?
$$f(r_i) =
begin{cases}
r_i^2, & text{if}~ |r_i| < 1.5 \
(1.5)^2 sgn^2(r_i), & text{if}~ |r_i| > 1.5
end{cases}$$
probability-distributions expected-value
asked Jan 26 at 13:11
neda neda
11
$endgroup$
$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47
add a comment |
$begingroup$
I have this discontinuous function:
$$f(r_i) =
begin{cases}
1, & text{if}~ |r_i| < 1.5 \
0, & text{if}~ |r_i| >1.5
end{cases}$$
where $r_i = z_i - a^2_ihat{x}~$, with $z_i$ being the $i$th column of matrix $z = Hx + e$, which error component $esim mathcal{N}(0, mathcal{I})$ has a covariance $mathcal{I}$ being the identity matrix, and $a_i$ is the $i$th column vector of a given matrix $H$.
I want to compute probability density function of it and then expectation value. How can I do that (calculating pdf function of these functions)?
What about this slightly different function below with the same $r_i$ ?
$$f(r_i) =
begin{cases}
r_i^2, & text{if}~ |r_i| < 1.5 \
(1.5)^2 sgn^2(r_i), & text{if}~ |r_i| > 1.5
end{cases}$$
probability-distributions expected-value
asked Jan 26 at 13:11
neda neda
11
$endgroup$
I have this discontinuous function:
$$f(r_i) =
begin{cases}
1, & text{if}~ |r_i| < 1.5 \
0, & text{if}~ |r_i| >1.5
end{cases}$$
where $r_i = z_i - a^2_ihat{x}~$, with $z_i$ being the $i$th column of matrix $z = Hx + e$, which error component $esim mathcal{N}(0, mathcal{I})$ has a covariance $mathcal{I}$ being the identity matrix, and $a_i$ is the $i$th column vector of a given matrix $H$.
I want to compute probability density function of it and then expectation value. How can I do that (calculating pdf function of these functions)?
What about this slightly different function below with the same $r_i$ ?
$$f(r_i) =
begin{cases}
r_i^2, & text{if}~ |r_i| < 1.5 \
(1.5)^2 sgn^2(r_i), & text{if}~ |r_i| > 1.5
end{cases}$$
probability-distributions expected-value
probability-distributions expected-value
asked Jan 26 at 13:11
neda neda
11
asked Jan 26 at 13:11
neda neda
11
edited Jan 29 at 15:12
neda
asked Jan 26 at 13:11
neda neda
11
asked Jan 26 at 13:11
neda neda
11
asked Jan 26 at 13:11
neda neda
11
11
$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47
add a comment |
$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47
$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47
add a comment |
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$begingroup$
I really don't know what your $sgn^2(r_i)$ is supposed to mean. If it's the sign (positive, negative) then squaring it makes it totally redundant.
$endgroup$
– Lee David Chung Lin
Jan 28 at 9:49
$begingroup$
I mean signum function that it is:sgn(ri)*sgn(ri)
$endgroup$
– neda
Jan 29 at 14:08
$begingroup$
I kept it because of 0 point. i thought it might be effective.
$endgroup$
– neda
Jan 29 at 14:44
$begingroup$
another point I should mention is that x_hat is calculated using iteratively reweighted least square
$endgroup$
– neda
Jan 29 at 14:47