Expected value of ratio of correlated random variables?

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?

Source : Link , Question Author : Jeff , Answer Author : Jeff

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