Expected value of ratio of correlated random variables?

For independent random variables α and β, is there a closed form expression for

E[αα2+β2]

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.

Answer

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,

E[αα2+β2]12P(α12)P(β214).

By Chebyshev’s inequality,

P(α12)=P(α112)P(|α1|12)=1P(|α1|12)14var(α)

By Markov’s inequality,

P(β214)=1P(β214)14E[β2]=14var(β)

Therefore,

E[αα2+β2]12(140.32)(140.32)>0.28

Is a more standard/systematic way to do what I’m doing here, that gets a tighter bound?

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

Leave a Comment