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3 Interval of definition By looking at an initial value problem dy/dx = f(x,y) with y(x0) = y0, it is not always possible to determine the domain of the solution y(x) or the interval over which the function y(x) satisfies the differential equation (a) Solve the equation dy/dx = xy3.

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Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutor. IÒ àPpI > ²õžHbÁå ¶¦ Yd‚/)*f õÿÓ!F *U à ›„Ó`Ÿti h"äÂK_C#Ò°Œ¹ëzd6ݬ4>j ÃËŠã ½Qr®,I6!KËß2ŽD"D»O)©SKÜAæ – ï\(Y’"BŠ ˜"ÿûRÄè Èé% mA^ $%¼ 0a—‘·í§ïk ¤åü IÁä Ô¬5KÅ—qCÁ ª2"d»˜À$ ìw»}óc‰HÄ HÏ ÄáøÄ $!. "discrete random variables and their probability distributions" to prove that 1) E(aYb) = aE(Y) b = aμ b 2) V(aYb) = a^2V(Y) = a^2 σ^2 Please help only if you can And if you need the theorems please let me know I will write them they are 4 theorems thanx alooooot best answer gets all the points.

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Þ5ø F}vªÿZ'$@²Çò£7GÀ?²ÇñªÂßR#Þ¹‰wä ÿM'±ßç޾ϗûêb¬pÜ’spŠ YûÍ f ÅÆ êÀªâÊç~;²~,?. Proof As we know, X and Y are independent if and only if fX;Y(x;y) = fX(x)fY(y) or, equivalently, fXjY(xjy)= fX(x) But then E(XjY =y)=åx xfXjY(xjy)=åx xfX(x)=E(X) 2 2 EE(g(X)jY)=E(g(X)) Proof Set Z = g(X) Statement (i) of Theorem 1 applies to any two rv’s Hence, applying it to Z and Y we obtain EE(ZjY)= E(Z) which is the same. Proposition 4 All sets with µ∗(E) = 0 are µ∗ measurable Proof If µ∗(E) = 0, then for an arbitrary set A⊂Xwe have µ∗(A) ≥µ∗(A\E) = µ∗(A\E) µ∗(A∩E) {z } 0, because A∩E⊂E Let M∗ be the class of all µ∗measurable sets Theorem 5 (Carath´eodory) M∗ is a σalgebra and µ∗ M∗ →0,∞ is a measure Proof We will split the proof into several steps.

In this *improvised* video, I show that if is a function such that f(xy) = f(x)f(y) and f'(0) exists, then f must either be e^(cx) or the zero function It'. F(x) = 1/x, c=3 I'm having trouble with this one I got it down to (3x)/(x3) but can't go further Could anyone help me out, please?. Má2 K 2 Ç !Ã 7c V' F ¥ ë8× µ É G!Ã %4 ` Û#Õ7c V' º v ± í ç ô º Ø!Ã 7c Ø0°6 6ä & _ X 8 Z.

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