18.01 Section, November 26, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Moving averages, probability
Review
• Moving average of f with window size h: (T f )(x) =
1
h
R x+h
x
f (u) du
• Angle sum formulas:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
R∞
• Probability density function: p(x) ≥ 0 such that −∞ p(x) dx = 1.
• Normal distribution: p(x) = s√12π e−
R∞
• Expectation: E = −∞ x p(x) dx
R∞
• Variance: −∞ (x − E)2 p(x) dx
√
• Standard deviation: Variance
(x−µ)2
2s2
with mean µ and standard deviation s
Problems
1. Let f (x) = sin x.
(a) What is the moving average of f with window size 2π?
(You shouldn’t have to compute any integrals.)
(b) Without computing any integrals, make a rough sketch of the moving average of f
with window size π.
(c) Now do an integral to compute (b) exactly.
1
(d) What happens when the window size gets really small?
(e) Back to a window size of π, what happens for f (x) = sin(200x)?
(You should be able to do this without integrating anything.)
2. Men’s heights are distributed normally with mean 70 in and standard deviation 3 in. Write
an integral expressing the percentage of men over 75 inches tall.
3. Write an integral expressing the standard deviation of the probability density function
(
1
sin(x) if 0 < x < π
p(x) = 2
0
otherwise.
4. Bonus question: Let g(x) be the function
(
100 if 1.99 < x < 2
g(x) =
0
otherwise.
Z
∞
f (t)g(t − x) dx is
Without integrating anything, describe what the function T (x) =
−∞
doing.
2