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
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Chapter2: Second-order Linear Odes
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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)
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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