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Let n be a positive integer and p and r two real numbers in the interval...

Let n be a positive integer and p and r two real numbers in the interval (0,1). Two random variables X and Y are defined on a the same sample space. All we know about them is that X∼Geom(p) and Y∼Bin(n,r). (In particular, we do not know whether X and Y are independent.) For each expectation below, decide whether it can be calculated with this information, and if it can, give its value (in terms of p, n, and r). Note that I’ve set this one not to give credit or show feedback until everything is correct, to make it harder to guess.

Is there enough information to compute it?

  1. E[X+Y]
  2. Is there enough information to compute it?
  3. E[XY]
  4. Is there enough information to compute it?
  5. E[X2+Y2]
  6. Is there enough information to compute it?
  7. E[(X+Y)2]   
  8. We start with the situation from Exercise 8.1: Let n be a positive integer and p and r two real numbers in the interval (0,1). Two random variables X and Y are defined on a the same sample space. We are given that X∼Geom(p) and Y∼Bin(n,r). Now, also assume that X and Y are independent, and compute each expectation below (in terms of p, n, and r).
  9. E[X+Y]
  10. E[XY]
  11. E[X2+Y2]
  12. E[(X+Y)2]
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