Answered: We call p(x, y) the probability density…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 31E

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Question

We call p(x, y) the probability density function for attributes x and y if the probability
that an arbitrary member of a population has attribute x over the interval (a, b) as well
as attribute y over the interval (c, d), corresponds to the volume under the graph of
p(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y)
over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y.

(a) Explain why the two conditions given at the end: “the total volume
under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y”
are essential in the context of probabilities. Explain in simple terms

(b) Express the probability (volume) referred to in the statement above as a
double integral. Intervals should correspond to the region
of integration and the order of integration are consistent and clearly indicated.
(c) Use answer in part (b) with
p(x, y) = (see image)

to find

1. Verify that p(x, y) satisfies the definition (given above) of a probability density
function.
2. the probability that x > 1/3. Show all steps
3. the probability that x < (1/3) + y. Show all steps.