Given the functions a(t), b(t) and c(t), let g(t, 1) a(t) g(t,2) b(t) 9(t,3) c(t) Let...
Let Ps have the inner product given by evaluation at -2, -1, 1, and 2. Let po(t)-1. P,()-t, and p20)- a. Compute the orthogonal projection of p2 onto the subspace spanned by Po and P1 b. Find a polynomial q that is orthogonal to Po and p,, such that Po P is an orthogonal basis for Span(Po P1, P2). Scale the polynomial q so that its vector of values at a2(Simplify your answer.) Let Ps have the inner product given...
Let X be a discrete random variable with PMF(a) Find P(X ≤ 9). (b) Find E[X] and Var(X). (c) Find MX(t), where t < ln 3.
5. Let X be a discrete random variable with the following PMF: for x = 0 Px(x)- for 1 for x = 2 0 otherwise a) Find Rx, the range of the random variable X. b) Find P(X21.5). c) Find P(0<X<2). d) Find P(X-0IX<2)
Let X be a discrete random variable, and let Y X (a) Assume that the PMF of X is Ka2 0 if x- -3, -2,-1,0,1,2,3 otherwise, where K is a suitable constant. Determine the value of K. (b) For the PMF of X given in part (a) calculate the PMF of Y (c) Give a general formula for the PMF of Y in terms of the PMF of X
Let X be a discrete random variable with probability function f(x). Prove that E[a + b g(X) + c h(X)] = a + bE[g(X)] + cE[h(X))], where g and h are functions, and a, b and c are constants.
X1 and X2 are discrete random variables (and are independent) with probability functions: p1(x1) = 1/3 for x1 = −2 ,−1 , 0 p2(x2) = 1/2 for x2 = 1, 6 Let Y = X1 + X2 a). Find the MGF of X1, X2 and Y b). USE MGFS derived in part a) to determine the probability mass function of Y. Note: MGFs in part a) must be used to determine the pmf of Y in part b)
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
3. (a) Let (X,Y) have the joint pmf (2 + y + k – 1)! P(X = 1, Y = y) => pip (1- P1 - p2), r!y!(k − 1)! where r, y=0,1,2, ..., k> 1 is an integer, 0 <P1 <1,0 <p2 <1, and p1 + P2 <1, find the marginal pmfs of X and Y and the conditional pmf of Y given X = r.
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...