i need help with finding chebyshev inequality, please

[david]

a random variable X has pdf


f(x)=(3/2)e^-|3x| -∞ < x < ∞


a. find the mean and variance of X
b. use the chebyshev inequality to find a number α such that (|X-μ_x>α)=.01


i try to integrate (3/2)xe^-|3x|dx from -∞ to ∞ but keep getting zero for the mean.


should i split -|3x| into -3x and 3x then integrate (3/2)e^-(3x) and (3/2)e^3x .

 

[daon2]

$\displaystyle xe^{-|3x|}$ is antisymmetric about $\displaystyle x=0$. So its integral will be 0 on symmetric intervals about $\displaystyle x=0$ (including the infinite, as we have convergence to 0). So $\displaystyle \mu$ is zero.

For the second one, let $\displaystyle k=\sqrt{10}$ (why?) and find $\displaystyle \alpha=k\sigma$. This gives you an inequality though, not equality.