4. Consider the set of all polynomials p(x, y) in two variables (x,y) € (0,1) (0,...
Consider the set of all polynomials of degree ≤3 in t defined over the interval [0,1]. (a) Find an orthonormal basis for this space. (b) Find the projection of the polynomial t^4 onto this space.
The set of polynomials p(x) = ax2 + bx + c that satisfy p(3) = 0 is a subspace of the vector space P2 of all polynomials of degree two or less. O True False
Exercise 4: Consider y= cos x over T0,1.2]. Determine the error bounds for the Lagrange polynomials P(), P(x) and P(x) Exercise 4: Consider y= cos x over T0,1.2]. Determine the error bounds for the Lagrange polynomials P(), P(x) and P(x)
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q. II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence class such that x ~ y (x is equivalent to y) if x-y є Q, the set of rational numbers 2. Given 1, by the Axiom of Choice, there exists a nonempty set B C [0,1) such that IB contains exactly one member of each equivalence class. Prove each of the following (a) Suppose that q E Qn [0, 1). Show that B (b)...
6. Let F be a field and a Fx] a nonconstant polynomial. Denote (that is, (a(x)) is the set of all polynomials in Flr] which are divisble by a()). Then (a) Prove that (a(x)) is a subgroup of the abelian group (Flx],. (b) consider the operation on F[r]/(a()) given by Prove that this operation is well-defined. (c) Prove that the quotient F]/(a(x) is a commutative ing with identity (d) What happens if the polynoial a() is constant? 6. Let F...
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P given by g) = Lif(t)g(x).rº dr. (a) Prove that (,) defines an inner product on P. (b) Let O be the set of odd polynomials, i.e. f(r) € P such that f(x)= -f(-r). Show that is a subspace of P. (c) Explain why g() = 5x2 - 3 is in 0+ (the orthogonal complement of O with respect to (>). (d) Let P<2 denote...
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...