Ex:           Find μX for the joint probability function shown below.

Sol'n:      We may express f(x, y) as a sum of two boxes:

We integrate in the y direction to find the marginal density function, fX(x), that we need in the calculation of μX.

We can calculate the above integrals, or we may observe that the values of the integrals represent the areas of cross sections in the y direction, as indicated in the following figure.

The areas are given by widths times heights.

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Now we integrate over x to find μX ≡ E(X).

or

or