3A4 (Minimizing penalty) Abel and Bertha are working in different weather forecasting com- panies. They have both calculated a predictive distribution for X, the Celsius temperature to- morrow at noon, and they agree that it has the triangular-shaped density function f(x) = 0.08 x over the interval [0, 5]. However, their companies and the general public won't hear anything about distributions. They want plain and simple point predictions. Abel has to pick a single point a € [0, 5] as his temperature prediction. Likewise Bertha has to pick a single point b € [0,5]. They can choose different points if they like. (a) Verify by integrating that f is really a continuous density function. (b) Find the mean μ = : E(X), and the median m, which is a point such that P(X ≤ m) = 1/. (c) Abel's company wants to encourage good predictions, so his salary is reduced by a quadratic penalty (X− a)², in some convenient units of money; where X is the tem- perature observed tomorrow, and a is the point Abel chose. Since X is not known yet, Abel is interested in his expected penalty q(a) = E ((X − a)²). Simplify this to obtain q as a simple polynomial function of a, not containing any E signs.
3A4 (Minimizing penalty) Abel and Bertha are working in different weather forecasting com- panies. They have both calculated a predictive distribution for X, the Celsius temperature to- morrow at noon, and they agree that it has the triangular-shaped density function f(x) = 0.08 x over the interval [0, 5]. However, their companies and the general public won't hear anything about distributions. They want plain and simple point predictions. Abel has to pick a single point a € [0, 5] as his temperature prediction. Likewise Bertha has to pick a single point b € [0,5]. They can choose different points if they like. (a) Verify by integrating that f is really a continuous density function. (b) Find the mean μ = : E(X), and the median m, which is a point such that P(X ≤ m) = 1/. (c) Abel's company wants to encourage good predictions, so his salary is reduced by a quadratic penalty (X− a)², in some convenient units of money; where X is the tem- perature observed tomorrow, and a is the point Abel chose. Since X is not known yet, Abel is interested in his expected penalty q(a) = E ((X − a)²). Simplify this to obtain q as a simple polynomial function of a, not containing any E signs.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![3A4 (Minimizing penalty) Abel and Bertha are working in different weather forecasting com-
panies. They have both calculated a predictive distribution for X, the Celsius temperature to-
morrow at noon, and they agree that it has the triangular-shaped density function f(x) = 0.08 x
over the interval [0, 5].
However, their companies and the general public won't hear anything about distributions.
They want plain and simple point predictions. Abel has to pick a single point a € [0,5] as his
temperature prediction. Likewise Bertha has to pick a single point b = [0,5]. They can choose
different points if they like.
(a) Verify by integrating that ƒ is really a continuous density function.
(b) Find the mean µ = = E(X), and the median m, which is a point such that P(X ≤ m) = ½.
(c) Abel's company wants to encourage good predictions, so his salary is reduced by a
quadratic penalty (X − a)², in some convenient units of money; where X is the tem-
perature observed tomorrow, and a is the point Abel chose. Since X is not known yet,
Abel is interested in his expected penalty q(a) = E ((X − a)²). Simplify this to obtain q
as a simple polynomial function of a, not containing any E signs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5de3a64f-a4ce-431a-a29f-75ac85a7201d%2Fbd7f0d85-aa46-427e-ba40-9c822bf9055e%2Fksn6d6c_processed.png&w=3840&q=75)
Transcribed Image Text:3A4 (Minimizing penalty) Abel and Bertha are working in different weather forecasting com-
panies. They have both calculated a predictive distribution for X, the Celsius temperature to-
morrow at noon, and they agree that it has the triangular-shaped density function f(x) = 0.08 x
over the interval [0, 5].
However, their companies and the general public won't hear anything about distributions.
They want plain and simple point predictions. Abel has to pick a single point a € [0,5] as his
temperature prediction. Likewise Bertha has to pick a single point b = [0,5]. They can choose
different points if they like.
(a) Verify by integrating that ƒ is really a continuous density function.
(b) Find the mean µ = = E(X), and the median m, which is a point such that P(X ≤ m) = ½.
(c) Abel's company wants to encourage good predictions, so his salary is reduced by a
quadratic penalty (X − a)², in some convenient units of money; where X is the tem-
perature observed tomorrow, and a is the point Abel chose. Since X is not known yet,
Abel is interested in his expected penalty q(a) = E ((X − a)²). Simplify this to obtain q
as a simple polynomial function of a, not containing any E signs.
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