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[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 216) (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((v1)*y)*dy 0 After applying partial integration etc. I end up with:
E(Y*exp(v*Y)) = 1/(1v)^2 for v < 1.
But according to Proakis the result is 2*v/((v+1)^2*(v1)^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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