(1) Let P denote the solid bounded by the surface of the hemisphere zV1--y2 and the cone z-Vx2 + y2 and let n denote an outwardly directed unit normal vector. Define the vector field (a) Evaluate the surface integral F nds directly without using Gauss' Divergence T heorem (b) Evaluate thetriplengral IIdiv(F) dV directly without using Gauss Diver- gence Theorem. confirming the result of Gauss' Divergence Theorem for this particular example. (1) Let P denote the solid bounded by the...
1.2 Let Yi and Y2 be independent random variables with Yi N(0, 1) and Y2 N(3,4). (a) What is the distribution of Y?? (b) If y-l (Y2-3)/2 | , obtain an expression for уту. What is its Yi and its distribution is yMVN(u, V), obtain an expression for yTV-ly. What is its distribution?
-Yi+ 6. Let F = Fii+F23 = 3. 202 + y2 22 + y² (a) Let C denote the unit circle x2 + y2 = 1 oriented counterclockwise. Calculate ScFodř without using Green's Theorem. That is, parameterize C and evaluate this line integral directly.
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....
6.62. Let Yi < Y2 < . . . < Y, be the order statistics of a random sample of size n from the distribution having p.df.f(x)-2x/g, 0<x <θ, zero elsewhere (a) If 0 < c < 1, show that Pr (c < Y,/θ < 1)-1-eM (b) If n=5 and if the observed value of Y, is 1.8, find a 99 percent confidence interval for 0.
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
12. (8 Pts.) Let Xi and X2 have the joint PDF Let Yi Xi/X2 and Y2 = Xy. Find the joint PDF of(H.)a). Are Y1 and Y2 independent?
Let C1 be the semicircle given by z = 0,y ≥ 0,x2 + y2 = 1 and C2 the semicircle given by y = 0,z ≥ 0,x2 +z2 = 1. Let C be the closed curve formed by C1 and C2. Let F = hy + 2y2,2x + 4xy + 6z2,3x + eyi. a) Draw the curve C. Choose an orientation of C and mark it clearly on the picture. b) Use Stokes’s theorem to compute the line integral ZC...
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a) Show that θ = y is an unbiased estimator for θ (b) Show that θ = 1Y is a minimum-variance estimator for θ.
A2) Let Sl be the unit circle z2 + y2-l in R2. Let S2 be the unit sphere z2 + y2 + z2-l in R. Let Sn be the unit hypersphere x| + z +--+ z2+1-1 in Rn+1 (a) Write an iterated double integral in rectangular coordinates that expresses the area inside S1. Write an iterated triple integral in rectangular coordinates that expresses the volume inside S2. Write an iterated quadruple integral in rectangular coordinates that expresses the hypervolume inside...