9.Consider the joint pdf3f(xr, y) =(4.xy + x²) for 0 < x < 1 and 1

Answered: Consider the joint pdf 3 f (x, y) =…

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Consider the joint pdf 3 f (x, y) = (4.xy + x²) for 0

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 77E

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Question

9.
Consider the joint pdf
3
f(xr, y) =
(4.xy + x²) for 0 < x < 1 and 1 <y< 2
10
otherwise
What is E(10XY)?
A. 10.458
В. 1
С. 8.5
D. 14.958
E. None of the above
10.
Given the joint pmf
p(x, y)
Y
1
.10 .04
.02
1
.08 .20
.06
2
.06 .14
.30
What is the probability x + y < 1?
A. .10
В. .78
С. .22
D. .42
E. None of the above

Expert Solution

Step 1

9.)

$E (10 X Y) = 10 E (X Y) E (X Y) = \int_{0}^{1} \int_{1}^{2} x y \frac{3}{10} (4 x y + x^{2}) d y d x = \int_{0}^{1} \frac{3 x}{10} [\int_{1}^{2} 4 x y^{2} d y + \int_{1}^{2} x^{2} y d y] d x = \int_{0}^{1} \frac{3 x}{10} [{[4 x \frac{y^{3}}{3}]}_{1}^{2} + {[x^{2} \frac{y^{2}}{2}]}_{1}^{2}] d x = \int_{0}^{1} \frac{3 x}{10} [\frac{4 x}{3} \times 7 + \frac{3 x^{2}}{2}] d x = \int_{0}^{1} \frac{28}{10} x^{2} d x + \int_{0}^{1} \frac{9}{20} x^{3} d x = \frac{28}{10} \times \frac{{[x^{3}]}_{0}^{1}}{3} + \frac{9}{20} \times \frac{{[x^{4}]}_{0}^{1}}{4} = \frac{28}{30} + \frac{9}{80} = \frac{1}{10} \times (\frac{28}{3} + \frac{9}{8}) = \frac{1}{10} \times \frac{224 + 27}{24} = \frac{1}{10} \times \frac{251}{24} H e n c e, E (10 X Y) = \frac{251}{24} = 10.458 (o p t i o n a)$

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