Re: Partly observed variable. Looks like a Bernoulli problem.
- From: Aniko <aniko123_57@xxxxxxxxx>
- Date: Fri, 06 Jul 2007 19:57:21 -0000
On Jul 5, 9:18 am, englishinpa...@xxxxxxx wrote:
I am struggling with the following problem. Can anybody help?
Let X be a positive bounded random variable with density f on the
range [0,M]. Some process generates a sample X1,X2,.. of iid
observations from f.
However you do not see all observations. Instead, you see Xi with
probability p Xi, where p is a fixed number between 0 and 1/M. With
probability 1-pXi you do not see Xi. In fact, you do not even know
that it has been generated. Let Y1, Y2... be the random variables that
you do see.
a)What is the density of Yi?
b)Express E[Yi] in terms of the moments of X and p.
Many thanks for your help.
Cute problem!
Here are two possible approaches:
1) Intuitive approach: since X=x occurs with probability f(x) and then
(independently) is observed with probability px, the density of Y will
be proportional to pxf(x). Just find the appropriate scaling constant
to make it a proper density.
2) Formal approach: go through the cumulative distribution function.
Note that having a Y value implies that it has been observed
P(Y<=y) = P(Y<=y|Y is observed) = P(Y<=y|X is observed) = P(Y<=y and X
is observed)/P(X is observed) = \int_0^y f(x)px dx / \int_0^M
pxf(x)dx.
I'll leave the working out of the details to you, just in case it was
a homework problem. But feel free to ask further questions if you get
stuck.
Aniko
.
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