[數學] Expectation

johnny138new

2010-05-09 9:50 am

A continuous random variable X has a Rayleigh distribution if and only if its probability density function is

f(x)= 2kxe^(-kx^2), for x>0
0 elsewhere

where k>0 is a parameter.

a) Find E(X). (Hint: Use the propeties of the gamma function)
b) Prove that E(X^2)=1/k

How to do these two questions?
Thank you!

回答 (1)

天助

2010-05-10 2:38 am

✔ 最佳答案

Lemma: Γ(a)=∫[0~∞] x^(a-1) e^(-x) dx (sub. x=t^2)
=2∫[0~∞] t^(2a-1) e^(-t^2) dt

Qa:
E(x)=∫[0~∞] xf(x)dx=2k ∫[0~∞] x^2 e^(-kx^2)dx (sub. t=√k x)
=2k∫[0~∞] (t^2 /k) e^(-t) dt/√k= (2/√k)∫[0~∞] t^2 e^(-t^2)dt
=(1/√k) *Γ(3/2) (by lemma: a=3/2)
= 0.5√(π/k) (Γ(3/2)=0.5Γ(1/2)=0.5√π )
Qb:
E(x^2)=∫[0~∞] x^2 f(x)dx=2k ∫[0~∞] x^3 e^(-kx^2)dx (sub. t=√k x)
=2k∫[0~∞] t^3/(k√k) e^(-t) dt/√k= (2/k)∫[0~∞] t^3 e^(-t^2)dt
=(1/k) *Γ(2) (by lemma: a=2)
=(1/k)*1!= 1/k (Γ(2)=1!=1)

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