An upper bound on the expected value of the square of random variable dominated by a geometric random variableLet X and Y be two random variables such that:1. 0 ≤ X ≤ Y2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).I would be grateful for any help of how one could upperbound E ( X 2 ) in terms of p.

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An upper bound on the expected value of the square of random variable dominated by a geometric random variableLet X and Y be two random variables such that:1.   0  ≤  X  ≤  Y2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).I would be grateful for any help of how one could upperbound       E    (      X    2    ) in terms of p.

Zoe Andersen · Accepted Answer

Step 1If   0  ≤  X  ≤  Y almost surely, then   0  ≤      X    2    ≤      Y    2   almost surely, as   x  ↦      x    2   is monotonous on             R        +  . We also know, that if   X  ≤  Y almost surely, then   E  X  ≤  E  Y. Thus   0  ≤  E  (      X    2    )  ≤  E  (      Y    2    )  =  (  E  Y      )    2    +  V  a  r  (  Y  )  =      1    p    +            1      −      p              p      2        =      1          p      2       Step 2Thus   E  (      X    2    )  ∈  [  0  ;      1          p      2        ]. And the is the best possible bound.

Let X and Y be two random variables such that: 1. 0 le X le Y. 2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).

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