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Chapter 3 Solutions
Pearson eText for Probability and Statistical Inference -- Instant Access (Pearson+)
- Show that if ax2+bx+c=0 for all x, then a=b=c=0.arrow_forwardProve that Var(E(Y|X)) ≤ Var(Y)arrow_forwardDetermine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)arrow_forward
- Let f1, f2, . fn be n real-valued, differentiable functions.- ... Prove that e d fifz .. fn] = (if …* fn) + (fifz… fn) + ……· + (fifz…*fi). - fn] = (fif2. fn) + (fif2 …· fn) + ...+ (fif2 fn). - dx (You may cite basic calculus theorems without proving them.)<arrow_forwardSuppose we have the quadratic function f(x)=A(x^2)+2X+C where the random variables A and C have densities fA(x)=(x/2) for 0≤x≤2, and fC(x)=3(x^2) for 0≤x≤1. Assume A and C are independent. Find the probability that f(x) has real rooarrow_forwardShow that E[Var(Y |X)] ≤ Var(Y )arrow_forward
- Show that if gcd(a, y) = 1 and a \ xy Then a\ a %3Darrow_forwardAre the following sets linear independent: (a) {p(x)=3x+2x², q(x) = −2+x+2x², r(x) = x + x²} (b) {u1 (1,2,3), u₂ = (3, 2, 1), uz = (0, 4, 8)} = Which of the following sets span R³ (a) S = {(1,1,0), (0, 1, 1)} (b) S = {(1,0, 1), (0, 1, 2), (−1, −4, 2)} Check if the set S is basis for the given vector space V. (a) S = {1, x, x², 2x² + x − 2} and V = P₂. -1 (b) S = {S₁ = ( 2 − 2) こ - , S2 = (32) }, and V = M2,2.arrow_forward(a) Use the rules of expected value to show that Cov(ax + b, cY + d) = ac Cov(X, Y). Cov(ax + b, CY + d) = E - E(ax + b)E ])-) + adx + bcY + bd - (aE(X) + b) CE = = acE + adE(X) + bcE(Y) + bd + adE(X) + bcE = acE acE(X)E 1)) = — асCov(X, Y) (b) Use part (a) along with the rules of variance and standard deviation to show that Corr(ax + b, cY + d) = Corr(X, Y) when a and c have the same sign. co( b, cY + Corr(ax + b, cY + d) = Oax + bº cY + d Cov(X, Y) la||clox0y Corr(X, Y) lallc| = Corr(X, Y) (c) What happens if a and c have opposite signs? ac O When a and c differ in sign, ac is negative and = -1. Therefore, Corr(ax + b, CY + d) = -Corr(X, Y). la||c| ac O When a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). Jal|c ас O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lal|c ac O when a and c differ in sign, ac is positive and = -1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). la||c| ас O…arrow_forward
- 3. Determine whether the Mean Value Theorem can be applied to f (x) = Vx + 3 on [-3,6]. Find all values of c in (-3,6) such that f'(c) = f(b)- f(a) , if possible. b-a Your answerarrow_forwardWhat is the relative risk for the association between men and women vs COVID-19 infection? What do you conclude?arrow_forward3. Determine a and b for Y X1 and Y2aX bX2 such that EX1 EX2}= 0, Var(X) Var(X2) Var(Yi) Var(2) 1, Cov(X, X2) 0, and Cov(Y, Y2) 0.5arrow_forward
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Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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