2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1. f(t) = :e This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt b. A related function is 2 F(x) = Var e9 dt, r> 0. This function gives the probability that the day's temperature is between t = -r and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the monthly average. (c) Now approximate your answer to (b) to within 0.001 of the actual value. Make sure you justify that the error in your approximation is no greater than 0.001.
2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1. f(t) = :e This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt b. A related function is 2 F(x) = Var e9 dt, r> 0. This function gives the probability that the day's temperature is between t = -r and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the monthly average. (c) Now approximate your answer to (b) to within 0.001 of the actual value. Make sure you justify that the error in your approximation is no greater than 0.001.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:2. In probability, it is common to model the deviation of a day's temperature from the monthly
average temperature using the Gaussian probability density function,
1.-/9
f(t) =
V9T
This means that the probability that the day's temperature will be between t = a and t = b
different from the monthly average temperature is given by the area under the graph of
y = f(t) between t a andt = b.
A related function is
%3D
F(x) =
-2/9 dt, 2 0.
%3D
V9n Jo
This function gives the probability that the day's temperature is between t = -x and t = x
different from the monthly average temperature. For example, F(1) 0.36 indicates that
there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1
degree less and 1 degree more) of the monthly average.
1
(a) Find a power series representation of F(x) (write down the power series using sigma
notation).
(b) Use your answer to (a) to find a series equal to the probability that the day's temperature
will be within 2 degrees of the monthly average.
(c) Now approximate your answer to (b) to within 0.001 of the actual value. Make sure you
justify that the error in your approximation is no greater than 0.001.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
