Derivation of moments of normalized B splines
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I am reading Ramsey's paper on Monotone splines https://projecteuclid.org/euclid.ss/1177012761 .
On page 3 he specifies that each $M_{i}$ has a properties of a probability density function over the interval $[t_{i},t_{i+k}]$. In general the random variable $X$ having distribution $M_{i}left(x | k,tright)$ has moments such that
$$Eleft[X ,big|~M_{i}(x, | k,t)right] = frac{ t_i + cdots + t_{i+k} }{k+1} \
Vleft[X ,big|~M_{i}(x, | k,t)right] = sum_{j=l+1}^{i+k} sum_{l=i}^{i+k} frac{ (t_{j}-t_{l})^2 }{ (k+1)^2(k+2) }$$
How do we derive this general formula for the mean and variance ? It is not very obvious to me. Any reference is appreciated.
probability probability-theory probability-distributions spline
asked Dec 31 '18 at 0:32
user24318user24318
427
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add a comment |
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I am reading Ramsey's paper on Monotone splines https://projecteuclid.org/euclid.ss/1177012761 .
On page 3 he specifies that each $M_{i}$ has a properties of a probability density function over the interval $[t_{i},t_{i+k}]$. In general the random variable $X$ having distribution $M_{i}left(x | k,tright)$ has moments such that
$$Eleft[X ,big|~M_{i}(x, | k,t)right] = frac{ t_i + cdots + t_{i+k} }{k+1} \
Vleft[X ,big|~M_{i}(x, | k,t)right] = sum_{j=l+1}^{i+k} sum_{l=i}^{i+k} frac{ (t_{j}-t_{l})^2 }{ (k+1)^2(k+2) }$$
How do we derive this general formula for the mean and variance ? It is not very obvious to me. Any reference is appreciated.
probability probability-theory probability-distributions spline
asked Dec 31 '18 at 0:32
user24318user24318
427
$endgroup$
add a comment |
$begingroup$
I am reading Ramsey's paper on Monotone splines https://projecteuclid.org/euclid.ss/1177012761 .
On page 3 he specifies that each $M_{i}$ has a properties of a probability density function over the interval $[t_{i},t_{i+k}]$. In general the random variable $X$ having distribution $M_{i}left(x | k,tright)$ has moments such that
$$Eleft[X ,big|~M_{i}(x, | k,t)right] = frac{ t_i + cdots + t_{i+k} }{k+1} \
Vleft[X ,big|~M_{i}(x, | k,t)right] = sum_{j=l+1}^{i+k} sum_{l=i}^{i+k} frac{ (t_{j}-t_{l})^2 }{ (k+1)^2(k+2) }$$
How do we derive this general formula for the mean and variance ? It is not very obvious to me. Any reference is appreciated.
probability probability-theory probability-distributions spline
asked Dec 31 '18 at 0:32
user24318user24318
427
$endgroup$
I am reading Ramsey's paper on Monotone splines https://projecteuclid.org/euclid.ss/1177012761 .
On page 3 he specifies that each $M_{i}$ has a properties of a probability density function over the interval $[t_{i},t_{i+k}]$. In general the random variable $X$ having distribution $M_{i}left(x | k,tright)$ has moments such that
$$Eleft[X ,big|~M_{i}(x, | k,t)right] = frac{ t_i + cdots + t_{i+k} }{k+1} \
Vleft[X ,big|~M_{i}(x, | k,t)right] = sum_{j=l+1}^{i+k} sum_{l=i}^{i+k} frac{ (t_{j}-t_{l})^2 }{ (k+1)^2(k+2) }$$
How do we derive this general formula for the mean and variance ? It is not very obvious to me. Any reference is appreciated.
probability probability-theory probability-distributions spline
probability probability-theory probability-distributions spline
asked Dec 31 '18 at 0:32
user24318user24318
427
asked Dec 31 '18 at 0:32
user24318user24318
427
edited Jan 2 at 18:46
user24318
asked Dec 31 '18 at 0:32
user24318user24318
427
asked Dec 31 '18 at 0:32
user24318user24318
427
asked Dec 31 '18 at 0:32
user24318user24318
427
427
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