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Integration of Joint Probability Density Functions

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Provide an example of how to set up the integral ranges to find a particular probability area.

Let Y_1 and Y_2 have joint probability density function given by

f(y_1,y_2) = 6(1-y_2) if 0 <= y_1 <= y_2 <= 1
0 elsewhere

a) Find P(Y_1 <= 3/4 , Y_2 >= 1/2)
b) Find P(Y_1 <= 3/4 , Y_2 <= 3/4)

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Solution Preview

The easiest way to set up integral ranges is to actually draw a diagram of the domain of the function. Note that the domain is bounded by the square 0 <= y_1 <= 1, 0 <= y_2 <=1. So let's start by drawing that. Then because there's an additional constraint that y_1 <= y_2, you'll want to draw a diagonal line y_1 = y_2, and shade the upper triangle. (If you're ever doubtful regarding which region to shade, pick a random point in each region, and test to see if that point satisfies ...

Solution Summary

The easiest way to set up integral ranges is to actually draw a diagram of the domain of the function. From your graph, determine whether the region of integration is simple. If not, methodically divide the region into parts that are simple.

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