A machine generates rectangular tiles with randomly varying length (L) and width (W). Let the length follow L~Gamma(100,10) distribution and the width follow W~Gamma(50,10) distribution, and define the area of the tile as S = L x W, and its perimeter as Q = 2(L+W). For parts (a-c) below, assume the length and width are independent (i.e. LLW). (a) Show that the expected value of the area is E[S] = 50. (b) Show that the variance of the area is Var[S] = 75.5. (c) Use Chebyshev's inequality to find an upper bound on P(S > 75). For parts (d,e) below, assume the length and width are correlated with Corr(L, W) = -0.5. (d) Show that the expected value of the area is E[S] ≈ 49.64645. (e) Find the mean and variance of the perimeter: E[Q] and Var[Q].
A machine generates rectangular tiles with randomly varying length (L) and width (W). Let the length follow L~Gamma(100,10) distribution and the width follow W~Gamma(50,10) distribution, and define the area of the tile as S = L x W, and its perimeter as Q = 2(L+W). For parts (a-c) below, assume the length and width are independent (i.e. LLW). (a) Show that the expected value of the area is E[S] = 50. (b) Show that the variance of the area is Var[S] = 75.5. (c) Use Chebyshev's inequality to find an upper bound on P(S > 75). For parts (d,e) below, assume the length and width are correlated with Corr(L, W) = -0.5. (d) Show that the expected value of the area is E[S] ≈ 49.64645. (e) Find the mean and variance of the perimeter: E[Q] and Var[Q].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 93E
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Question
![A machine generates rectangular tiles with randomly varying length (L) and width (W). Let
the length follow L ~Gamma(100,10) distribution and the width follow W ~Gamma(50,10)
distribution, and define the area of the tile as S = L × W, and its perimeter as Q = 2(L+W).
For parts (a-c) below, assume the length and width are independent (i.e. LI W).
(a) Show that the expected value of the area is E[S] = 50.
(b) Show that the variance of the area is Var[S] = 75.5.
(c)
Use Chebyshev's inequality to find an upper bound on P(S > 75).
For parts (d,e) below, assume the length and width are correlated with Corr(L, W)
= -0.5.
(d) Show that the expected value of the area is E[S] ~ 49.64645.
(e) Find the mean and variance of the perimeter: E[Q] and Var[Q].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12f51b64-c3b4-40b6-b565-16b387e10bf1%2F400d7400-aeee-4371-ba44-2fe7fba08731%2Fbxa83zi_processed.png&w=3840&q=75)
Transcribed Image Text:A machine generates rectangular tiles with randomly varying length (L) and width (W). Let
the length follow L ~Gamma(100,10) distribution and the width follow W ~Gamma(50,10)
distribution, and define the area of the tile as S = L × W, and its perimeter as Q = 2(L+W).
For parts (a-c) below, assume the length and width are independent (i.e. LI W).
(a) Show that the expected value of the area is E[S] = 50.
(b) Show that the variance of the area is Var[S] = 75.5.
(c)
Use Chebyshev's inequality to find an upper bound on P(S > 75).
For parts (d,e) below, assume the length and width are correlated with Corr(L, W)
= -0.5.
(d) Show that the expected value of the area is E[S] ~ 49.64645.
(e) Find the mean and variance of the perimeter: E[Q] and Var[Q].
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