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In this exercise, do not attempt formal mathematical derivations, which would actually involve some subtle issues when we go beyond discrete random variables. Rather, use your understanding of the concepts involved. For each one of the statements below, indicate whether it is true or false.
(a) The law of iterated expectations tells us that E [E[X|Y]] = E[X]. Suppose that we want apply this law in a conditional universe, given another random variable Z, in order to evaluate E [X2]. Then: EE[X|Y, 2]|2] = E[X2] y E[E[X|Y]|2] =E[X2] V EE[X|Y,Z]] =E[X2]
(b) Determine whether each of the following statements about the quantity E[g(X,Y)|Y,Z) is true or false. The quantity E[9(X,Y)|Y, 2) is: • a random variable y a number y a function of (X,Y) y a function of (Y,Z) | a function of Z only

Sagot :

Solution :

From the given equation :

E[ E (X|Y) ] = E (X)

a). Then,

   E[ E [ X|Y,Z] | Z] = E [ X|Z ]

     ----  True

   E [ E [ X|Y ] | Z ] = E [ X|Z ]

    ---- False

  E [E [X | Y,Z ]] = E [X|Y ]

    ----  False

b). Th quantity E [ g (X,Y) | Y,Z ] is ,

  • A random variable  -----  True
  • A number  ----- False
  • A function of (X,Y)  -----   False
  • A function of (Y,Z) -----   True
  • A function of Z only  -------   False

The low of iteration tell the following statement are true E[ E [ X|Y,Z] | Z] = E [ X|Z ] . A random variable y . A function of (Y,Z)

From the given equation the law of iterated expectations

[tex]E[ E (X|Y) ] = E (X)[/tex]

Therefore We have to find a)

What is the definition of iteration?

Iteration is the repetition of a process in order to generate a sequence of outcomes.

So by using the low of iteration we can say that,

E[ E [ X|Y,Z] | Z] = E [ X|Z ]     ----  True

E [ E [ X|Y ] | Z ] = E [ X|Z ]  ---- False

E [E [X | Y,Z ]] = E [X|Y ]   ----  False

b). Th quantity E [ g (X,Y) | Y,Z ] is ,

For a random variable y this is  -----  True

For a number ----- False

For a function of (X,Y)  -----   False

For a function of (Y,Z) -----   True

For function of Z only  -------   False

Therefore,The low of iteration tell the following  statement are true E[ E [ X|Y,Z] | Z] = E [ X|Z ] . A random variable y . A function of (Y,Z)

To learn more about the iteration visit:

https://brainly.com/question/14284157

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