(d) Show that if and are distinct eigenvalues of a square matrix A, x = (x1; x2; : : : ; xn) 2 E[A], y = (y1; y2; : : : ; yn) 2 E[A] then: x; y = x1y1 + x2y2 + + xnyn = 0:
(d) Show that if and are distinct eigenvalues of a square matrix A, x = (x1; x2; : : : ; xn) 2 E[...
(d) Show that if λ and µ are distinct eigenvalues of a square matrix A, x = (x1, x2, . . . , xn) ∈ Eλ[A], y = (y1, y2, . . . , yn) ∈ Eµ[A] then: x, y = x1y1 + x2y2 + · · · + xnyn = 0
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
(a) If C is the line segment connecting the point (X1,Y1) to the point (X2, y2), find the following. Jexdy-y x dy - y dx O A = (b) If the vertices of a polygon, in counterclockwise order, are (X1,Y1), (x2, y2), ..., (Xn, Yn), find the area of the polygon. O A = 3 [(x112 - – *287) + (x3X3 – x3y2) + ... + (*n – 1'n – XnYn – 1) + (xn/1 – xqYn] = {[(x112 +...
(a) If C is the line segment connecting the point (X1,Y1) to the point (X2, y2), find the following. e x dyr dy - y dx xly2 - x2y1 x A= A= (b) If the vertices of a polygon, in counterclockwise order, are (X1,Y1). (X2, y2), ..., (X, Yn), find the area of the polygon. [0x271 – 1/2) + (x392 – x2Y3) + .. + ... + (xnxn-1 - xn-1n) + (*11n – Xnxx)] + x2+1) + (x2y + x372)...
3.7 Problems 3.1 Show that (a) the product on R2 defined by (x1, x2)(y1, y2) = (x1y1 – x2y2, x1 y2 + x2yı) (b) turns R2 into an associative and commutative algebra, and the cross product on R3 turns it into a nonassociative, noncommutative algebra.
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
(a) Show that the points (x1, yı), (X2, y2), ..., (xn, yn) are collinear in R2 if and only if 1 X1 yi X2 Y3 rank < 2 1 Xn yn ] (b) What is the generalization of part (a) to points (x1, Yı, zı), (x2, y2, 22), ...,(Xn, Yn, zn) in R'. Explain.
Consider a random sample (X1, Y1),(X2, Y2), . . . ,(Xn, Yn) where Y | X = x is modeled by a N(β0 + βx, σ2 ) distribution, where β0, β1 and σ 2 are unknown. (a) Prove that the mle of β1 is an unbiased estimator of β1. (b) Prove that the mle of β0 is an unbiased estimator of β0.
Suppose that X1, X2,.... Xn and Y1, Y2,.... Yn are independent random samples from populations with the same mean μ and variances σ., and σ2, respectively. That is, x, ~N(μ, σ ) y, ~ N(μ, σ ) 2X + 3Y Show that is a consistent estimator of μ.