Solution :
E(aX) = axi pi= a xi pi= a E(X)
By using definition ,E(X) = xi pi
from the formula E(aX+b)=aE(x)+b, setting b = 0 we see that E(aX)= aE(X) Prove E(aX) =...
1. Prove the following results and give the logical explanation. a. E(aX) = aE (X), where a is a constant. b. E(X + b) = E(X) + b, where b is a constant. You are interested in studying the distribution of the workout days of 100 students. The 2. collected data is summarized in the table below: Workout Days 15 18 20 24 Students Count 10 17 33 20 12 8 Calculate the average number of workout days for students?...
Let X be a metric space and let E C X. The boundary aE of E is defined by E EnE (a) Prove that DE = E\ E°. Here Eo is the set of all interior points of E; E° is called the interior of E (b) Prove that E is open if and only if EnaE Ø. (c) Prove that E is closed if and only if aE C E (d) For X R find Q (e) For X...
Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that E(Y) = aE(X) b, and Var(X)a2Var(X). What about the density of Y, fy(y)? Assuming a > 0. Calculate fy(y) using the following two methods (1) Let Fx() P(X x). Calculate Fy(y) = P(Y < y) in terms of Fx. Then calculate fy (2) Calculate Y (y, y + Ay)) Ay fr(y) (3) Give geometric explanations of your result
Determine e At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula eAt = X(t)X(0)1. 0 2 2 2 0 2 2 2 0 First, find X(t). Choose the correct answer below 4t -2t 4t e -2t (1+t) e e -2t OA. X(t) (1+t)e4t 0 e2t B. X(t)= e4t 0 -2t -2t 2t - e - 2t (1t)e 4t e e 4t e 4t - sint sin t 0 (1t)e -2t O C....
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
(1) Let (, A, i) be a measure space. {AnE: Ae A} is a o-algebra of E, contained in (a) Fix E E A. Prove that Ap = A. (b) Let uE be the restriction of u to AĘ. Prove that iE is a measure on Ag. (c) Suppose that f : Q -» R* is measurable (with respect to A). Let g = the restriction of f to E. Prove that g : E ->R* is measurable (with respect...
a) Prove that for all x, y≥0 we have |√x−√y|≤√|x-y|. (b) Prove that f(x)=√x is uniformly continuous on [0,∞).
(1) Let (, A, /i) be a measure space = {AnE: A E A} is a o-algebra of E, contained in (a) Fix E E A. Prove that AE A. (b) Let be the restriction of u to AE. Prove that uE is a measure on Ag (c) Suppose that f -> R* is measurable (with respect to A). Let g = f\e be the restriction of f to E. Prove that g E ->R* is measurable (with respect to...
Assume a, b ∈ Z. Prove that if ax + by = 1 for some x, y ∈ Z, then gcd(a, b) = 1.
Solve the following system (X'=AX) using the formula: A= X(t) = 4x We were unable to transcribe this image