Expected value of X = E(X) = ∑xP(x) ∑xf
1 96 0 95 0 01 1 96 0 02 cm Px x P PZ PZ μ μ σσ σ σ σ −≤ = ⎛− ⎞ ⎜⎟≤= ⎝⎠ ⎛⎞ ⎜⎟≤= ⎝⎠
8 Fourier Series
n=1 [an cosnx+bn sinnx] the Fourier series of f We denote this fact by f(x) ∼ a0 2 + X∞ n=1 [an cosnx+bn sinnx] The symbol ∼ should be read as f “has Fourier serier” Example 8 1 Let f be a 2π-periodic function given by f(x) = x for x ∈ (−π,π] Then integrating by parts an = 1 π Z π −π xcosnx dx = 1 π xsinnx n π − 1
Lecture 3 : The Natural Exponential Function: ) =
Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = ex Last day, we saw that the function f(x) = lnxis one-to-one, with domain (0;1) and range (1 ;1) We can conclude that f(x) has an inverse function f 1(x) = exp(x) which we call the natural exponential function The de nition of inverse functions gives us the following:
1 Definition and Properties of the Exp Function
e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N and n 6= 0 Then erx = e m n x = e 1 n x m = (e ) m = (e ) m = (e )r Derivatives Lemma 4 • d dx ex = ex ⇒ Z ex dx = ex +C • dm dxm e x= ex > 0 ⇒ E(x) = e is concave up, increasing, and positive Proof Since E(x) = ex is the inverse of L(x) = lnx, then with y
LECTURE 8: Continuous random variables and probability
• Definit·on o variance: var(X) - E[(X - µ) 2 1 • Calcu ation us·ng he expected value rule, E g(X)] - 1-:g(x)fx(x) dx var(X) Standard deviation: ax = var(X) var(a b) = a2var(X) A useful formula: var(X) - E x2 - ( [X] 2
510 Taylor and Maclaurin Series - Brian Veitch
5 10 Taylor and Maclaurin Series Brian E Veitch c 4 = f4(a) 4 3 2 It appears that if f(x) has a power series representation, then c n = fn(a) n The next theorem will pretty much state the same thing, but a bit more formally
Math 361, Problem set 6 - University of Denver
1 (1 8 3) Let X have pdf f(x) = (x+2)=18 for 2 < x < 4, zero elsewhere Find E[X];E[(X + 2)3] and E[6X 2(X + 2)3] Answer: E[X] = 1 18 Z 4 22 x2 + 2xdx = 1 18 (x3 4
Solutions to HW5 Problem 31
ECE302 Spring 2006 HW5 Solutions February 21, 2006 3 Problem 3 2 1 • The random variable X has probability density function fX (x) = ˆ cx 0 ≤ x ≤ 2, 0 otherwise
AP Calculus Name Pd Product & Quotient Rules D(1) Day 4 1 2
Product & Quotient Rules D(1) Day 4 1-10: Differentiate 1 e) 2 x 2 e() x 3 12 34 x gx x 4 2 2 21 x Gx x 5 u() ¨¸ 6 3 24 13 y5 yy §· ©¹ 7 2 42 2 31 t y tt 8 p p 9 1 3 tt gt t 10 1 x x e y e
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