Probability
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ngeva0
Guest
Posted: Wed Dec 14, 2005 9:15 am    Post subject: Probability Reply with quote
x[n] is a zero-mean white Gaussian random process with variance of 1 an
y[n] is the output when x[n] is filtered using a two-tap FIR filter wit
coefficients of [1 1]. What's the probability of the event {y[n+1]>1
given that y[n]=1?
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Jerry Avins
Guest
Posted: Wed Dec 14, 2005 9:15 am    Post subject: Re: Probability homework? Reply with quote
ngeva0 wrote:
Quote:
x[n] is a zero-mean white Gaussian random process with variance of 1 and
y[n] is the output when x[n] is filtered using a two-tap FIR filter with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

Is the noise bandlimited before filtering? What is the effect of your
filter on any signal? (Given y[n] = a and y[n+1] = b, what will the
filter output be?)

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
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Stan Pawlukiewicz
Guest
Posted: Wed Dec 14, 2005 5:16 pm    Post subject: Re: Probability Reply with quote
Stan Pawlukiewicz wrote:
Quote:
ngeva0 wrote:

x[n] is a zero-mean white Gaussian random process with variance of 1 and
y[n] is the output when x[n] is filtered using a two-tap FIR filter with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

Technically the probability that y(n)=1, is zero for a continuous
density. In practice it isn't. If y(n) is governed by a discrete
probability density, the question makes some more sense.


Btw, I know that finals are happening all over the place. I wouldn't
normally answer this sort of question at this time of year, but I have a
thing about a professors that ask trick questions. IMHO they are chixen
fhit pricks.
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Stan Pawlukiewicz
Guest
Posted: Wed Dec 14, 2005 5:16 pm    Post subject: Re: Probability Reply with quote
ngeva0 wrote:
Quote:
x[n] is a zero-mean white Gaussian random process with variance of 1 and
y[n] is the output when x[n] is filtered using a two-tap FIR filter with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

Technically the probability that y(n)=1, is zero for a continuous

density. In practice it isn't. If y(n) is governed by a discrete
probability density, the question makes some more sense.
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ngeva0
Guest
Posted: Wed Dec 14, 2005 11:42 pm    Post subject: Re: Probability homework? Reply with quote
Quote:
ngeva0 wrote:
x[n] is a zero-mean white Gaussian random process with variance of
and
y[n] is the output when x[n] is filtered using a two-tap FIR filte
with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

Is the noise bandlimited before filtering? What is the effect of your
filter on any signal? (Given y[n] = a and y[n+1] = b, what will the
filter output be?)

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ


the output y[n]=x[n]+x[n-1] after filtering. therefore, y[n+1]=x[n+1]+x[n]
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Randy Yates
Guest
Posted: Wed Dec 14, 2005 11:56 pm    Post subject: Re: Probability Reply with quote
Got some karma on this one, do you Stan??? :)

--RY
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Stan Pawlukiewicz
Guest
Posted: Thu Dec 15, 2005 12:03 am    Post subject: Re: Probability Reply with quote
Randy Yates wrote:
Quote:
Got some karma on this one, do you Stan??? :)

--RY

All I can say is that it's good that I went to college before I learned

to use C4 in the Army.
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Guest
Posted: Thu Dec 15, 2005 12:11 am    Post subject: Re: Probability Reply with quote
Yup, I agreee....this a tricky problem in Random processes....related
to linear systems, characteristic funtions. This particular operation
would correlate the uncorrelated input signal. So, to find the pdf of
the output is your problem...It has been a long time for me dealing
with this stuff....but I hope my 'clues' are correct

Nithin
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Stan Pawlukiewicz
Guest
Posted: Thu Dec 15, 2005 12:29 am    Post subject: Re: Probability Reply with quote
nithin.pal@gmail.com wrote:
Quote:
Yup, I agreee....this a tricky problem in Random processes....related
to linear systems, characteristic funtions. This particular operation
would correlate the uncorrelated input signal. So, to find the pdf of
the output is your problem...It has been a long time for me dealing
with this stuff....but I hope my 'clues' are correct

Nithin



P( y(n)=1 ) = 0 , characteristic functions don't change that. Either it
is a typo, or a trick question.
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Jerry Avins
Guest
Posted: Thu Dec 15, 2005 12:33 am    Post subject: Re: Probability homework? Reply with quote
ngeva0 wrote:
Quote:
ngeva0 wrote:

x[n] is a zero-mean white Gaussian random process with variance of 1

and

y[n] is the output when x[n] is filtered using a two-tap FIR filter

with

coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

Is the noise bandlimited before filtering? What is the effect of your
filter on any signal? (Given y[n] = a and y[n+1] = b, what will the
filter output be?)

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ



the output y[n]=x[n]+x[n-1] after filtering. therefore, y[n+1]=x[n+1]+x[n]

Since the input to the filter isn't bandlimited, there will be aliasing
at any sample rate, so beware of intuitive results. Nevertheless, if
y[n] = 1, then x[n] + x[n-1] = n. You need to calculate the probability
that x[n+1] will exceed x[n-1]. That's not trivial, but at least it's
not a conditional probability. Pick a (Gaussian) number. Pick another.
What is the probability that the second is more positive than the first?

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
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Stan Pawlukiewicz
Guest
Posted: Thu Dec 15, 2005 12:41 am    Post subject: Re: Probability homework? Reply with quote
Jerry Avins wrote:
Quote:
ngeva0 wrote:

ngeva0 wrote:

x[n] is a zero-mean white Gaussian random process with variance of 1


and

y[n] is the output when x[n] is filtered using a two-tap FIR filter


with

coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?


Is the noise bandlimited before filtering? What is the effect of your
filter on any signal? (Given y[n] = a and y[n+1] = b, what will the
filter output be?)

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ



the output y[n]=x[n]+x[n-1] after filtering. therefore,
y[n+1]=x[n+1]+x[n]


Since the input to the filter isn't bandlimited, there will be aliasing
at any sample rate, so beware of intuitive results. Nevertheless, if
y[n] = 1, then x[n] + x[n-1] = n. You need to calculate the probability
that x[n+1] will exceed x[n-1]. That's not trivial, but at least it's
not a conditional probability. Pick a (Gaussian) number. Pick another.
What is the probability that the second is more positive than the first?

Jerry
Jerry - Aliasing has nothing to do with the problem.
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Guest
Posted: Thu Dec 15, 2005 12:48 am    Post subject: Re: Probability Reply with quote
y[n] = 1
=> x[n-1] + x[n] = 1
=> x[n] = 1 - x[n-1]

P{ y[n+1]>1 }
= P{ (x[n+1]+x[n]) > 1 }
= P{ (x[n+1]+1-x[n-1]) > 1 }
= P{ (x[n+1]-x[n-1]) > 0 }
= P{ x[n+1]>x[n-1] }

Since x[n] has a symmetrical distribution

P{ x[n+1]>x[n-1] } = P{ x[n+1]<x[n-1] } = 1/2



ngeva0 wrote:
Quote:
x[n] is a zero-mean white Gaussian random process with variance of 1 and
y[n] is the output when x[n] is filtered using a two-tap FIR filter with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?
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Stan Pawlukiewicz
Guest
Posted: Thu Dec 15, 2005 1:00 am    Post subject: Re: Probability Reply with quote
kd_ei@yahoo.com wrote:
Quote:
y[n] = 1
=> x[n-1] + x[n] = 1

but P( x(n-1) + x(n) = 1 ) = 0


Quote:
=> x[n] = 1 - x[n-1]

P{ y[n+1]>1 }
= P{ (x[n+1]+x[n]) > 1 }
= P{ (x[n+1]+1-x[n-1]) > 1 }
= P{ (x[n+1]-x[n-1]) > 0 }
= P{ x[n+1]>x[n-1] }

Since x[n] has a symmetrical distribution

P{ x[n+1]>x[n-1] } = P{ x[n+1]<x[n-1] } = 1/2



ngeva0 wrote:

x[n] is a zero-mean white Gaussian random process with variance of 1 and
y[n] is the output when x[n] is filtered using a two-tap FIR filter with
coefficients of [1 1]. What's the probability of the event {y[n+1]>1}
given that y[n]=1?

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Jerry Avins
Guest
Posted: Thu Dec 15, 2005 1:06 am    Post subject: Re: Probability homework? Reply with quote
Stan Pawlukiewicz wrote:

...

Quote:
Jerry - Aliasing has nothing to do with the problem.

I wrote that it not needing to be considered, it wouldn't interfere with
any insights. My error was writing that it's not a conditional probably
problem. It is, but not in the way I had in mind.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
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Guest
Posted: Thu Dec 15, 2005 1:17 am    Post subject: Re: Probability Reply with quote
Jerry Avins write:
....
Quote:
The probability that a random number lies within some interval is finite
and decreases as the interval gets smaller. In the limit, when the width
of the interval goes to zero -- i.e., becomes a single number -- the
probability [goes to] [becomes] [is] (pick one) zero.

That depends completely on the probability density. If it has a point
mass at that number, then the probability is not equal to zero.
Consider the following probability distribution on the intervall
[0,1], defined by

P[ x \in [0, 1/2[ ] = 1/3
P[ x = 1/2 ] = 1/3
P[ x \in ]1/2, 1] ] = 1/3.

It has positive probability for x being equal to 1/2.

Regards,
Andor
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