Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / M / 3 / a

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- Solution a

Solution

The marginal probability density function is

f_{Y} (y) = \int_{R} f_{X, Y} (x, y) d x

. For

y

with

f_{Y} (y) > 0

, the conditional density is

f_{X ∣ Y = y} (x) = \frac{f _{X, Y} ( x , y )}{f _{Y} ( y )} .

If the conditional first absolute moment is finite, the conditional expectation is

E [X ∣ Y = y] = \int_{R} x \frac{f _{X, Y} ( x , y )}{f _{Y} ( y )} d x .

Conditioning here uses a density, since the individual event

Y = y

normally has probability zero; a conditional law is specified only up to marginal-null values of

y

.

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