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Topic: [Mathqa]Calculating Chernoff bound for Laplace distribution
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 Ruben Lysens Posts: 1 Registered: 12/20/04
[Mathqa]Calculating Chernoff bound for Laplace distribution
Posted: Mar 24, 2001 6:45 AM

Hi,

I've got a question about an example in Proakis' book 'Digital
Communications' (Example 2-1-6) (not off topic. This is about
probability, not Digital Communications) :

I'd like to calculate the Chernoff bound for a Laplace distributed
random variable:

The Laplace pdf: p(y) = (1/2)*exp(-|y|)

The Chernoff bound for the upper tail probability: P(Y>=delta) >=
exp(-v*delta)*E(exp(v*Y))

Where v is the solution to the following equation: E(Y*exp(v*Y)) -
delta*E(exp(v*Y)) = 0

So, to solve the above equation I need to find the moments E(Y*exp(v*Y))
and E(exp(v*Y)) for the given pdf.

All this you can find in Proakis' book. From here on I'm on my own, so
tell me where I'm going wrong:

I'm not used to posting to math groups. I don't know how you write your
math symbols in ASCII (feel free to fill me in), so I'll use my own
symbols:

oo
{f(x)*dx means: integrate f(x) for x going from 0 to positive infinity.
o

For the upper tail: y >= 0

oo
E(Y*exp(v*Y)) = {y*exp(v*y)*(1/2)*exp(-y)*dy
0
oo
= {(1/2)*y*exp((v-1)*y)*dy
0

After applying partial integration etc. I end up with:

E(Y*exp(v*Y)) = 1/(1-v)^2 for v < 1.

But according to Proakis the result is 2*v/((v+1)^2*(v-1)^2)

Solving the integral was relatively straightforward. I think I got that
one right, so I probably started out with the
wrong formula, or wrong bounds or something like that.

I won't go into E(exp(v*Y)), I'd just make the same mistake.

Can anybody help me out here?

Thanks,

Ruben.

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