Areas under normal distribution curves can be used calculate probabilities of many random variables in the natural and social sciences. The graph of the Standard Normal Curve, with a mean of 0 and a standard deviation of 1 is given below, and its function is given by 1 f(x) = √2π 1.5 1 The integral of L e da would be used to calculate the probability of an event that occurs √2π 1.5 between -1.5 to 1.5 standard deviations from the mean. 1 Unfortunately, e 2 dx has no proper closed form antiderivative, so we must use numerical √2TT 1.5 methods to approximate the values. (-1.5, 0.13) 1.5 -115 (-1,0.24) (-0.5, 0.35) Using Simpson's Rule, -0.5 04 -0-2 [ 15 e- 3 dr ~ [ е 0 (0, 0.4) 015 Using the Midpoint Rule with three rectangles of equal width, Using the Trapezoidal Rule with six trapezoids of equal width, 1.5 ·[e- (0.5, 0.35) 1.5 [tbe е (1, 0.24) da ≈ 115 (1.5, 0.13)
Areas under normal distribution curves can be used calculate probabilities of many random variables in the natural and social sciences. The graph of the Standard Normal Curve, with a mean of 0 and a standard deviation of 1 is given below, and its function is given by 1 f(x) = √2π 1.5 1 The integral of L e da would be used to calculate the probability of an event that occurs √2π 1.5 between -1.5 to 1.5 standard deviations from the mean. 1 Unfortunately, e 2 dx has no proper closed form antiderivative, so we must use numerical √2TT 1.5 methods to approximate the values. (-1.5, 0.13) 1.5 -115 (-1,0.24) (-0.5, 0.35) Using Simpson's Rule, -0.5 04 -0-2 [ 15 e- 3 dr ~ [ е 0 (0, 0.4) 015 Using the Midpoint Rule with three rectangles of equal width, Using the Trapezoidal Rule with six trapezoids of equal width, 1.5 ·[e- (0.5, 0.35) 1.5 [tbe е (1, 0.24) da ≈ 115 (1.5, 0.13)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Areas under normal distribution curves can be used calculate probabilities of many random variables in the
natural and social sciences.
The graph of the Standard Normal Curve, with a mean of 0 and a standard deviation of 1 is given below,
and its function is given by
1
f(x) =
√2T
e
1
The integral of
e dx would be used to calculate the probability of an event that occurs
√2T
-1.5
between -1.5 to 1.5 standard deviations from the mean.
(-1.5, 0.13)
1
Unfortunately,
√2π -1.5
methods to approximate the values.
1.5
-1.5
1.5
(-1, 0.24)
Using Simpson's Rule,
(-0.5, 0.35)
e dx has no proper closed form antiderivative, so we must use numerical
1.5
-1.5
1-0.5
-0-2
Using the Midpoint Rule with three rectangles of equal width,
dx
0
Using the Trapezoidal Rule with six trapezoids of equal width,
(0, 0.4)
0.5
Round answers to 4 decimal places when appropriate.
(0.5, 0.35)
(1, 0.24)
1.5
Ibe
1.5
1.5
[ 13 e-t de = \
е
1.5
e-2
е dx
(1.5, 0.13)
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