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