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eimsori [14]
3 years ago
7

For this problem, prove/derive the formula that allows one to find an expected value for X by conditioning on Y :

Mathematics
1 answer:
zimovet [89]3 years ago
6 0

Answer:

E[X |Y=y] = \int_{-\infty}^{\infty} x f_{X,Y} (x|y) dx

E[X|Y=y] =\sum_{x} x f_{X,Y} (x|y)

Step-by-step explanation:

For this case we assume that we have two random variable X and Y continuous, and we define the conditional density of X given Y like this:

f_{X|Y} (x|y) = \frac{f_{X,Y} (x,y)}{f_Y (y)}

Where f_{X,Y} (x,y) is the joint density function. And we can define the conditional probability like this:

P(a\leq X \leq b | Y = y) = \int_{a}^b f_{X,Y} (x|y) dx

In order to find the expected value of X given Y=y we just need to find this:

E[X | Y=y] = \int_{-\infty}^{\infty} x f_{X,Y} (x|y) dx

And if we assume that the random variable is discrete then the conditional expectation would be given by:

E[X|Y=y] =\sum_{x} x f_{X,Y} (x|y)

And as we can se just change the integral by a sum over the values defined for X, and with this we have the general formulas in order to find the conditional expectation of X given Y=y for the possible cases for a random variable.  

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