Re: Partly observed variable. Looks like a Bernoulli problem.

On Jul 9, 5:03 pm, Aniko <aniko123...@xxxxxxxxx> wrote:
On Jul 9, 2:38 pm, englishinpa...@xxxxxxx wrote:



On Jul 9, 3:31 pm, englishinpa...@xxxxxxx wrote:

On Jul 6, 3:57 pm, Aniko <aniko123...@xxxxxxxxx> wrote:

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

Hello Aniko,

Thank you very much! I really appreciate your help.

I think I understand the numerator and denominator you wrote. That
problem was easy for you it looks like. I need a lot of practice on
those distributional problems.

Sincerely
Michel

I would be grateful if you could answer the following question.

Was it possible to solve that problem using a partial distribution
function f approach instead of the cumulative distribution function F
approach?

I understand that your method is good and that all I need to do is
differentiate your F in order to obtain the f. I was just wondering if
there might be another approach.

Many thanks

Michel,

I am not sure what exactly do you mean by a "partial distribution
function approach". What I called the intuitive approach does use only
the pdf f. I am sure it could be formalized somehow. If you are
thinking of using the chain-rule like formula for finding the pdf of
Y=g(X), I don't see how could that be used directly, since there is no
nice "g" function that would describe the transformation. It is not
even a real transformation, since some of the X's disappear.

Aniko

Again many thanks for your help Aniko. I was confused and I now
realize that my question does not make sense.

Could you give me a hint on how to solve question b)

b)Express E[Yi] in terms of the moments of X and p.

I have tried to use
EY=E{E[Y|X]}
But I am not getting anything interesting.

.

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