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) = V97 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 and t = b. A related function is 2 F(x) = t2 /9 dt, x > 0. 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) 2 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) = V97 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 and t = b. A related function is 2 F(x) = t2 /9 dt, x > 0. 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) 2 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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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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
f(t) =
V97
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 and t = b.
A related function is
2
F(x) =
t2 /9 dt, x > 0.
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) 2 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.
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