How could you show that the Nakagami density integrates to 1?
https://en.wikipedia.org/wiki/Nakagami_distribution
If X is a Nakagami random variable with parameters A and B, then Y = sqrt(X) is a Gamma with parameters A, and B/A. You can show that P(X < k) = P(Y^2 < k) = P(Y < sqrt(k)), and using the Gamma distribution, taking k to infinity, do you prove that F(X) tends to 1.
In the Gamma distribution is easy to see that lim_{k to infinity} F(k) = 1 because you will have an integral divided by the Gamma function, but when k goes to infinity the integral is the Gamma function, then lim_{k to infinity} F(k) = Gamma(A, B/A) / Gamma(A, B/A) = 1.
Alternatively, integrate $x^{2k-1] exp(-x^2)$ by parts and use induction. A fairly straightforward early integral calculus exercise.
1 Recommendation
11th Nov, 2015
Pelumi Oguntunde
Covenant University Ota Ogun State, Nigeria
@ Calkin, thank you for the information
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