b******e
发帖数: 118 | 1 1. option1: $240 fixed payment
option2: 25% chance to win $1000
which one do you choose? What if the prob is 40%?
General question: 3 variables, value of fixed payment, probability and
corresponding payment(i.e. 3 numbers mentioned above), generate an algorithm
to make the decision to choose either one.
2. Two independent random variables, x--N(0,sigmax^2), y--N(0,sigmay^2),
where sigmax and sigmay are known, you have one sample from x and one sample
from y, but you don't know the exact | P****d
发帖数: 113 | 2 1. depending on your utility function.
for example, if your u(x)=x^p (0
then just compare E[U(X)]=prob*(corresponding payment)^p
and fixed payment.
2.MLE: joint likelihood function
L_{sigmax,sigmay }(X,Y) is proportional to
exp{-0.5*(X/sigmax)^2-0.5*(Y/sigmay)^2},
s.t. X+Y=Z.
the result is X=Z* sigmax^2/(sigmax^2+sigmay^2).
algorithm
sample
the
【在 b******e 的大作中提到】
: 1. option1: $240 fixed payment
: option2: 25% chance to win $1000
: which one do you choose? What if the prob is 40%?
: General question: 3 variables, value of fixed payment, probability and
: corresponding payment(i.e. 3 numbers mentioned above), generate an algorithm
: to make the decision to choose either one.
: 2. Two independent random variables, x--N(0,sigmax^2), y--N(0,sigmay^2),
: where sigmax and sigmay are known, you have one sample from x and one sample
: from y, but you don't know the exact
| b******e
发帖数: 118 | 3 How do you get the result Z* sigmax^2/(sigmax^2+sigmay^2)? Could you please
explain it in detail?
Btw, for the utility function you mentioned in question 1, is there a
general way to determine its form, i.e. quadratic, exponential or whatever?
Thanks a lot!
【在 P****d 的大作中提到】
: 1. depending on your utility function.
: for example, if your u(x)=x^p (0: then just compare E[U(X)]=prob*(corresponding payment)^p
: and fixed payment.
: 2.MLE: joint likelihood function
: L_{sigmax,sigmay }(X,Y) is proportional to
: exp{-0.5*(X/sigmax)^2-0.5*(Y/sigmay)^2},
: s.t. X+Y=Z.
: the result is X=Z* sigmax^2/(sigmax^2+sigmay^2).
:
| S*******D
发帖数: 47 | 4 For the second question, you can follow the orthogonal principle like this:
assume xhat = h*z
then e = x - xhat = x - h*z
By O.P.
E[e*z]=0
then
sigmax^2 = h*(sigmax^2+sigmay^2).
then
h = sigmax^2/(sigmax^2+sigmay^2).
Does it make sense?
algorithm
sample
the
【在 b******e 的大作中提到】
: 1. option1: $240 fixed payment
: option2: 25% chance to win $1000
: which one do you choose? What if the prob is 40%?
: General question: 3 variables, value of fixed payment, probability and
: corresponding payment(i.e. 3 numbers mentioned above), generate an algorithm
: to make the decision to choose either one.
: 2. Two independent random variables, x--N(0,sigmax^2), y--N(0,sigmay^2),
: where sigmax and sigmay are known, you have one sample from x and one sample
: from y, but you don't know the exact
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