i need help with finding chebyshev inequality, please

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 .

\(\displaystyle xe^{-|3x|}\) 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.