Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density 1 0 < x < y < 1 f(x,y) y = elsewhere (a) Determine if X and Y are independent. (b) Find P(X + Y > 1/2). Figure 0.1: Hint1: the blue area is 0 < x < y < 1 and the green area is x+y> 1/2. The overlapping region (domain of integration) is a quadrilateral. It is a little bit hard to integrate. Y 0 / X Figure 0.2: Hint2: the blue area is 0 < x < y < 1 and the green area is x + y < 1/2. The overlapping region (domain of integration) is a triangle. У 0
Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density 1 0 < x < y < 1 f(x,y) y = elsewhere (a) Determine if X and Y are independent. (b) Find P(X + Y > 1/2). Figure 0.1: Hint1: the blue area is 0 < x < y < 1 and the green area is x+y> 1/2. The overlapping region (domain of integration) is a quadrilateral. It is a little bit hard to integrate. Y 0 / X Figure 0.2: Hint2: the blue area is 0 < x < y < 1 and the green area is x + y < 1/2. The overlapping region (domain of integration) is a triangle. У 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let X denote the diameter of an armored electric cable and Y denote the diameter
of the ceramic mold that makes the cable. Both X and Y are scaled so that they
range between 0 and 1. Suppose that X and Y have the joint density
Transcribed Image Text:1
0 < x < y < 1
f(x,y)
y
=
elsewhere
(a) Determine if X and Y are independent.
(b) Find P(X + Y > 1/2).
Figure 0.1: Hint1: the blue area is 0 < x < y < 1 and the green area is x+y> 1/2. The
overlapping region (domain of integration) is a quadrilateral. It is a little bit
hard to integrate.
Y
0<xy<1
x+y> /
X
Figure 0.2: Hint2: the blue area is 0 < x < y < 1 and the green area is x + y < 1/2. The
overlapping region (domain of integration) is a triangle.
У
0<xy<
x + y</
x
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