Solution: ?Suppose that (X, Y) is the outcome of an experiment that must occur in a

Chapter 5, Problem 118

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Suppose that (X, Y) is the outcome of an experiment that must occur in a particular region S in the xy -plane. In this context, the region S is called the sample space of the experiment and X and Y are random variables. If D is a region included in S, then the probability of (X, Y) being in D is defined as \(P[(X, Y) \in D]=\iint_{D} p(x, y) d x \ d y\), where p(x, y) is the joint probability density of the experiment. Here, p(x, y) is a nonnegative function for which \(\iint_{S} p(x, y) d x \ d y=1\).

Assume that a point (X, Y) is chosen arbitrarily in the square \([0,3] \times[0,3]\) with the probability density \(p(x, y)=\left\{\begin{array}{l}\frac{1}{9}(x, y) \in[0,3] \times[0,3] \\0 \text { otherwise }\end{array}\right.\).

Find the probability that the point (X, Y) is inside the unit square and interpret the result.

Text Transcription:

P[(X, Y) in D] = iint_D p(x, y) dx dy

iint_S p(x, y) dx dy = 1

[0,3] times [0,3]

p(x, y) = l 1/9 (x, y) in [0,3] times [0,3] over 0 otherwise