For independent random variables α and β, is there a closed form expression for
in terms of the expected values and variances of α and β? If not, is there a good lower bound on that expectation?
Update: I may as well mention that E[α]=1 and E[β]=0. I can control the variance on α and β, and I have in mind a setting where the variances of both α and β are pretty small relative to E[α]. Maybe both of their standard deviations are less than 0.3.
I thought of one lower bound, though I don’t think it’s very tight. I just pick an arbitrary value less than the mean of α and another arbitrary value around the mean of β2. Since the expectation is of a non-negative random variable, and because α and β are independent,
By Chebyshev’s inequality,
By Markov’s inequality,
Is a more standard/systematic way to do what I’m doing here, that gets a tighter bound?