. The normal curve (or the Gaussian), defined as g(x) = 27 plays an important role in probability and statistics. It is useful to know that | g(x) dx = 1, which we will now show. (a) Suppose that Da is a disk, centered at the origin, with radius a. Explain why *e-*+v*)/2 dr dy = lim // e-(z²+u*)/2 dA. (b) Change to polar coordinates to evaluate || e-(x²+y²)/2 dA, and take the limit as a –→+∞. (c) Explain why +oo e-(2²+y³)/2 dA = 1² /2 dx e-v³/2 dy e -00 (d) Conclude that e-2* /2 dx = V27, so that g(x) dx = 1, as we hoped.
. The normal curve (or the Gaussian), defined as g(x) = 27 plays an important role in probability and statistics. It is useful to know that | g(x) dx = 1, which we will now show. (a) Suppose that Da is a disk, centered at the origin, with radius a. Explain why *e-*+v*)/2 dr dy = lim // e-(z²+u*)/2 dA. (b) Change to polar coordinates to evaluate || e-(x²+y²)/2 dA, and take the limit as a –→+∞. (c) Explain why +oo e-(2²+y³)/2 dA = 1² /2 dx e-v³/2 dy e -00 (d) Conclude that e-2* /2 dx = V27, so that g(x) dx = 1, as we hoped.
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:3. The normal curve (or the Gaussian), defined as g(x)
plays an important role in probability and
V27
+oo
statistics. It is useful to know that
| g(x) dx = 1, which we will now show.
-00
(a) Suppose that Da is a disk, centered at the origin, with radius a. Explain why
+oo
e-(z²+y³)/2
dx dy
lim
e-(a²+y²)/2
dA.
(b) Change to polar coordinates to evaluate
e-(a2+y²)/2 dA, and take the limit as a → +o.
Da
(c) Explain why
too
+oo
+oo
too
"|e-=²+s²)/2 dA =
x² /2 dx
-y²/2
dy
-00
-00
(d) Conclude that
|e-a*/2 dx = V27, so that
dx = 1, as
we hoped.
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