n*******3
发帖数: 313 | 1 if X is a random variable with E(X), V(X). what is E(log X) and V(LogX)? |
b*****n
发帖数: 3774 | 2 你是问lognorma的expectation和variance哇，随便一本stat的书都有，
E是e^(mu+1/2*theta)
V是（e^theta^2-1)e^(2mu+theta^2)
太难打了。你google一哈嘛。
【在 n*******3 的大作中提到】
: if X is a random variable with E(X), V(X). what is E(log X) and V(LogX)?
|
j*******g
发帖数: 2140 | 3 what is the distribution of X?
【在 n*******3 的大作中提到】
: if X is a random variable with E(X), V(X). what is E(log X) and V(LogX)?
|
n*******3
发帖数: 313 | 4 you don't know the distribution of X, not necessarily normal.
【在 b*****n 的大作中提到】
: 你是问lognorma的expectation和variance哇，随便一本stat的书都有，
: E是e^(mu+1/2*theta)
: V是（e^theta^2-1)e^(2mu+theta^2)
: 太难打了。你google一哈嘛。
|
s****r
发帖数: 2386 | 5 If you don't know the distribution of X, then I guess the best you can get
is some bound (I mean some inequalities, through Jensen). |
n*******3
发帖数: 313 | 6 unknown dist of X. suppose density function f(X).
can we derive the analytical relationship between E(X) and E(logX), V(X) and
V(logX)?
【在 j*******g 的大作中提到】
: what is the distribution of X?
|
b*****n
发帖数: 3774 | 7 哦你要任何f(x）的分析解哇。
【在 n*******3 的大作中提到】
: you don't know the distribution of X, not necessarily normal.
|
A*L
发帖数: 2357 | 8 let me give it a shot
I think you may need to use the inverse function relation.
Assume log(x) = Y
To have E(Y), we need to have Y * {pdf of Y}
so now it is up to find pdf of Y.
exp(Y) = X
pdf of Y = (pdf of x) * (exp(Y))'
Now you get pdf of Y, you can get E and V.
Here is my help, and later please post your whole solution to let the
others, who may have interest, to read. Thank you!
【在 n*******3 的大作中提到】
: if X is a random variable with E(X), V(X). what is E(log X) and V(LogX)?
|
j*******g
发帖数: 2140 | 9 No. Depends on the distribution of X.
你必须要知道pdf of X 阿...
and
【在 n*******3 的大作中提到】
: unknown dist of X. suppose density function f(X).
: can we derive the analytical relationship between E(X) and E(logX), V(X) and
: V(logX)?
|
F***x
发帖数: 339 | 10 哇，川版真的是万能的。。。
【在 A*L 的大作中提到】
: let me give it a shot
: I think you may need to use the inverse function relation.
: Assume log(x) = Y
: To have E(Y), we need to have Y * {pdf of Y}
: so now it is up to find pdf of Y.
: exp(Y) = X
: pdf of Y = (pdf of x) * (exp(Y))'
: Now you get pdf of Y, you can get E and V.
: Here is my help, and later please post your whole solution to let the
: others, who may have interest, to read. Thank you!
|
|
|
j*******g
发帖数: 2140 | 11 显然她想知道能不能只用E(X), 和V(X), 就把E(log(X)) V(log(X))表达出来.
【在 A*L 的大作中提到】
: let me give it a shot
: I think you may need to use the inverse function relation.
: Assume log(x) = Y
: To have E(Y), we need to have Y * {pdf of Y}
: so now it is up to find pdf of Y.
: exp(Y) = X
: pdf of Y = (pdf of x) * (exp(Y))'
: Now you get pdf of Y, you can get E and V.
: Here is my help, and later please post your whole solution to let the
: others, who may have interest, to read. Thank you!
|
j*******g
发帖数: 2140 | 12 If you know the pdf of X, assuming Y=f=log(x),
Pdf of y = f inverse substituted in pdf of X * derivative of f inverse.
f inverse is e^y here. so derivative of f inversse is still e^y
so you can integrate y*pdf of y and get the expression of expectation.
Finally, you need to connect what you get with expectation of x and variance
of x (when you know the pdf of x of course)
【在 j*******g 的大作中提到】
: No. Depends on the distribution of X.
: 你必须要知道pdf of X 阿...
:
: and
|
n*******3
发帖数: 313 | 13 The problem is that you only know the E(X), pdf(x) is uncertain. Only way we
can solve E(log(X)) is to built the mathematical relationship between E(
logX) and E(X).
. how can we solve pdf of Y = (pdf of x) * (exp(Y))'.
【在 A*L 的大作中提到】
: let me give it a shot
: I think you may need to use the inverse function relation.
: Assume log(x) = Y
: To have E(Y), we need to have Y * {pdf of Y}
: so now it is up to find pdf of Y.
: exp(Y) = X
: pdf of Y = (pdf of x) * (exp(Y))'
: Now you get pdf of Y, you can get E and V.
: Here is my help, and later please post your whole solution to let the
: others, who may have interest, to read. Thank you!
|
j*******g
发帖数: 2140 | 14 you really need to know pdf of x...probably u can get pdf of x from the
expression of E(x) and V(X)..
we
【在 n*******3 的大作中提到】
: The problem is that you only know the E(X), pdf(x) is uncertain. Only way we
: can solve E(log(X)) is to built the mathematical relationship between E(
: logX) and E(X).
: . how can we solve pdf of Y = (pdf of x) * (exp(Y))'.
|
n*******3
发帖数: 313 | 15 No analytical relationship?
【在 j*******g 的大作中提到】
: No. Depends on the distribution of X.
: 你必须要知道pdf of X 阿...
:
: and
|
A*L
发帖数: 2357 | 16 can we just fool around? assuming E(x^2), E(x^3) ... are known too?
hahaha, then we could use taylor expansion .......
【在 n*******3 的大作中提到】
: No analytical relationship?
|
n*******3
发帖数: 313 | 17 lol..... good thoughts. admire~
I was thinking of using central limit theorem....
twisting....
【在 A*L 的大作中提到】
: can we just fool around? assuming E(x^2), E(x^3) ... are known too?
: hahaha, then we could use taylor expansion .......
|
A*L
发帖数: 2357 | 18 乱整了，下午间了哈
E'(x) = x * p(x) => p(x)
put into the notation below
E'(y) = p(x)*exp(x)
then you could get E(Y) expressed by E(x) ....
【在 A*L 的大作中提到】
: can we just fool around? assuming E(x^2), E(x^3) ... are known too?
: hahaha, then we could use taylor expansion .......
|
n*******3
发帖数: 313 | 19 u think it can be approximately derived by using Delta method?
【在 j*******g 的大作中提到】
: you really need to know pdf of x...probably u can get pdf of x from the
: expression of E(x) and V(X)..
:
: we
|
A*L
发帖数: 2357 | 20 check our what I just said above
【在 n*******3 的大作中提到】
: u think it can be approximately derived by using Delta method?
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|
o****o
发帖数: 8077 | 21 I think so
\sqrt(n)[X-E(X)] \stackrel{D}{\rightarrow} N{0, V(X)}
by Delta method
\sqrt{n} [log(X)-log(E(X))] \stackrel{D}{\rightarrow}N{0, V(X)/(E(X))^2}
so
log(X) has asymptotically approximated mean log(E(X)) and variance
\frac{V(X)}{n E(X)^2}
correct me if I am wrong
【在 n*******3 的大作中提到】
: u think it can be approximately derived by using Delta method?
|
j*******g
发帖数: 2140 | 22 VAR （log（X））那里，分母上有一个n？
CV of X 在某些条件下可以约等于log（X）的方差。 所以不太理解你那个n。
【在 o****o 的大作中提到】
: I think so
: \sqrt(n)[X-E(X)] \stackrel{D}{\rightarrow} N{0, V(X)}
: by Delta method
: \sqrt{n} [log(X)-log(E(X))] \stackrel{D}{\rightarrow}N{0, V(X)/(E(X))^2}
: so
: log(X) has asymptotically approximated mean log(E(X)) and variance
: \frac{V(X)}{n E(X)^2}
: correct me if I am wrong
|
o****o
发帖数: 8077 | 23 yes, it is from the \sqrt{n} on the LHS
【在 j*******g 的大作中提到】
: VAR （log（X））那里，分母上有一个n？
: CV of X 在某些条件下可以约等于log（X）的方差。 所以不太理解你那个n。
|
s****1
发帖数: 170 | 24 With X only being restricted by its mean and variance, E[\log{X}] and V(\log
{X}) could be ANYTHING. It's fairly easy to construct such examples.
【在 n*******3 的大作中提到】
: if X is a random variable with E(X), V(X). what is E(log X) and V(LogX)?
|
o****o
发帖数: 8077 | 25 en, right
say X~U(0, 1)
log
【在 s****1 的大作中提到】
: With X only being restricted by its mean and variance, E[\log{X}] and V(\log
: {X}) could be ANYTHING. It's fairly easy to construct such examples.
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