[問題] 求分配
有幾個求分配的題目想請教一下
1.Suppose that f(x,y)=1 for 0<x<1, 0<y<1 and =0 otherwise. Obtain f(x│X<Y).
f(x│X<Y) = f(x,x<y)/f(x<y)
請問f(x,x<y)該怎麼求呢?
2.Let X and Y be independent and distributed as N(μ,1) and as N(0,μ),
respectively, where μ>0. Derive the asymptotic variance of the maximum
likelihood estimator of μ based on seperate sample of X and Y and combined
sample (X1,...Xn , Y1,...,Yn).
seperate sample of X: _
maximum likelihood estimator of μ = X
_
Var(X) = σ^2/n
我不懂為什麼題目要加上一個asymptotic
和直接問variance of maximum likelihood estimator of μ有什麼不同?
3.Let X1,...,Xn be independent random variables, each with the exponential
distribution:P(X>=x)=e^(-αx), x>=0. Put X(n)=max{X1,...Xn}. and
bn=α^(-1)logn. What is the limiting distribution of X(n)-bn?
令Yn = max{X1-bn,...,Xn-bn}
Yn的cdf => F(y) = P(Yn=<y) = P[(X1-bn)=<y,...,(Xn-bn)=<y]
= [Fx(y+bn)]^n = {1-e^[-α(y+bn)]}^n
= {1-e^[-α(y+α^(-1)logn)]}^n
當n->∞,...我就卡住了,看不出來極限分配是什麼...?
4.Let the i.i.d. sequence{Xi} with pdf f(x)=2x^(-3), 1=<x<∞, Could we find
_
the probability limit of X?
_
我的想法是先找出X的pdf
_
X=(1/n)(X1+...+Xn)
我學過的Jocobian轉換法
好像沒辦法處裡這個問題
不知道該用哪一種方法比較好?
煩請指教~謝謝^^
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