Let $Y:= -2displaystylesum_{i=1}^n ln F_{X_i}(X_i)$. Prove that $Y$ have distribution $chi^2(2n)$
If $X_1,X_2,ldots,X_n$ be independent random variables each with distribution function $F_{X_i}$. Let $$Y:= -2sum_{i=1}^n ln F_{X_i}(X_i).$$
Prove that $Y$ have distribution $chi^2(2n)$.
I'm trying solve this problem, but I can't. Any hints or tip? Thanks for your help.
probability statistics
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
add a comment |
If $X_1,X_2,ldots,X_n$ be independent random variables each with distribution function $F_{X_i}$. Let $$Y:= -2sum_{i=1}^n ln F_{X_i}(X_i).$$
Prove that $Y$ have distribution $chi^2(2n)$.
I'm trying solve this problem, but I can't. Any hints or tip? Thanks for your help.
probability statistics
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53
add a comment |
If $X_1,X_2,ldots,X_n$ be independent random variables each with distribution function $F_{X_i}$. Let $$Y:= -2sum_{i=1}^n ln F_{X_i}(X_i).$$
Prove that $Y$ have distribution $chi^2(2n)$.
I'm trying solve this problem, but I can't. Any hints or tip? Thanks for your help.
probability statistics
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
If $X_1,X_2,ldots,X_n$ be independent random variables each with distribution function $F_{X_i}$. Let $$Y:= -2sum_{i=1}^n ln F_{X_i}(X_i).$$
Prove that $Y$ have distribution $chi^2(2n)$.
I'm trying solve this problem, but I can't. Any hints or tip? Thanks for your help.
probability statistics
probability statistics
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
edited Oct 17 '13 at 11:34
Stefan Hansen
20.7k73663
20.7k73663
asked Oct 17 '13 at 0:32
user63192
614411
asked Oct 17 '13 at 0:32
user63192
614411
asked Oct 17 '13 at 0:32
user63192
614411
614411
Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53
add a comment |
Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53
Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53
Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53
add a comment |
1 Answer
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Hints:
Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.
Show that $-2log(U)$ follows a $chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $chi^2(2)$ distribution is the same as an exponential distribution with parameter $tfrac12$.
Conclude using that $X+Ysimchi^2(n_1+n_2)$ if $Xsim chi^2(n_1)$ and $Ysimchi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hints:
Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.
Show that $-2log(U)$ follows a $chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $chi^2(2)$ distribution is the same as an exponential distribution with parameter $tfrac12$.
Conclude using that $X+Ysimchi^2(n_1+n_2)$ if $Xsim chi^2(n_1)$ and $Ysimchi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
add a comment |
Hints:
Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.
Show that $-2log(U)$ follows a $chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $chi^2(2)$ distribution is the same as an exponential distribution with parameter $tfrac12$.
Conclude using that $X+Ysimchi^2(n_1+n_2)$ if $Xsim chi^2(n_1)$ and $Ysimchi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
add a comment |
Hints:
Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.
Show that $-2log(U)$ follows a $chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $chi^2(2)$ distribution is the same as an exponential distribution with parameter $tfrac12$.
Conclude using that $X+Ysimchi^2(n_1+n_2)$ if $Xsim chi^2(n_1)$ and $Ysimchi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
Hints:
Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.
Show that $-2log(U)$ follows a $chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $chi^2(2)$ distribution is the same as an exponential distribution with parameter $tfrac12$.
Conclude using that $X+Ysimchi^2(n_1+n_2)$ if $Xsim chi^2(n_1)$ and $Ysimchi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
answered Oct 17 '13 at 11:32
Stefan Hansen
20.7k73663
20.7k73663
add a comment |
add a comment |
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Meeow - my mistake.
– wolfies
Oct 17 '13 at 11:53