Let W(t) be a Brownian motion. For each of the following processes, verify whether it is a martingale process:(1) X(t)=W^2(t)−t(2) X(t)=W^3(t)−3tW(t)(3) X(t)=t^2W(t)−2∫uW(u)du (The limit on the integral is 0 to t)(4) X(t)=W^2(t)
Let {W(t) : 0 <= t <= 1} be a Wiener process with W(0) = 0 and the parameter α2= 1 (such a process is called a standard Wiener process).
When answering the following questions be sure to justify your answers.
a.) Let X(t) = W(t +α) – W(t) for some α > 0. Is {X(t)} a Gaussian Process?
b.) Is {X(t)} a Wiener process?
c.) Is {X(t)} mean-square continuous?
d.) Let B(t) = W(t) – t*W(1) for 0 <= t <= 1. Is {B(t)} a Gaussian Process?
e.) Derive the covariance function for {B(t)}.
f.) Show that for any t, B(t) is independent of W(1).
Let V = span((1, 1, 0), (0, 0, 1)). Let prv : R3 → R3 denote the linear transformation whose value on a vector w ∈ R3 is equal to prv (w), the orthogonal projection of w onto V . Use the Dimension Theorem (Theorem 12.1 from the notes) to prove that the nullity of prv is equal to 1. And then find the standard matrix A of prv : R3 → R3.
Theorem 12.1 (Dimension Theorem). Let F : Rn → Rm be a linear transformation. Then
rank of F + nullity of F = n.
Hello, can you help me with the question above? Thanks!
5. Let V = P2 and define a pairing of poiynomials p and r; in [,2 by (19.9) == 1I(0)q(0) +p(1)r1(1) +p(2)q(2). (a) (7 points) Show that this formula defines an inner product on Pg. (1)) (4 points) Show that the polynomials p(t) = 4t2 — 8t + 2 and q(t) = —t2 — t + 4 areorthogonal with respect to this inner product. (0) (5 points) Find the matrix C that represents this inner product with respect to theordered basis S = (t2,t,1). (d) (4 points) Use coordinate vectors and your answer in part (c) to againshow that 4t2 — 8t + 2 and —t2 w t + 4 are orthogonal.
Let V be an inner product space and let W be a subspace of V . The orthogonal complement of W in V is the set W⊥ = v ∈ V | hv, wi = 0 for all w ∈ W .
Prove the following:
a) W⊥ is a subspace of V .
b) W⊥ ∩ W = {0}. That is, the only vector they have in common is the zero vector.
Hint: For the second statement, assume that there is a vector W not equal to 0 that is common to both. Show that this leads to something that we know to be impossible, and therefore such a W cannot exist. This method is called a proof by contradiction.
Please help me prove the following about sequences of continuous functions
Let X, Y be compact metric spaces and consider the metric space X x Y equippedwith the metric d((x1, y1), (x2, y2)) = dx(x1, 72) + dy (y1, y2). Prove that for everyf EC(X xY) and every e > 0, there exist functions gi, . . . On E C(X) and hi, . .. hn EC(Y) such thatnIf (x, y) -gi (x )hi(y) <e for all x EX, yEY.j=1
4. Let X be a continuous random variable, with CDF F(x), taking values in an interval[0, b]; that is, F(0) = 0 and F(b) = 1. Then there is an alternative formula forexpected value:E(X) =Z b0(1 − F(x)) dx. (1)(a) Assume b is a finite number. Prove (1) using integration-by-parts. [Hint:Recall that the PDF is f(x) = ddxF(x).](b)Check that the formula (1) holds when X Unif(0, b).(c) Formula (1) also works for b = 1. Check this when X is an Exponential RVwith PDF f(x) = e−x for x 0.
5. Let x = x(u, v) = u + v2, and y = y(u, v) = u2 + v4.(a). Find out where the Jacobian of the transformation equals 0, and find the chambers in the u-v plane where the Jacobian is positive and the chambers where it is negative.(b). Find the curves in the x-y plane that correspond to the rays in the u-v plane where the Jacobian equals 0. (One of the curves is only half a curve!)(c). As you go counterclockwise about the origin in the u-v plane, describe the correspondingmotion in the x-y plane, using your answer to part (a).
Let X bar be the mean of a random sample from the exponential distribution.
a) show that xbar is unbiasedpoint estimator of θ
b) using the mgf technique,determine the distribution of xbar
c) use (b) to show that Y= 2nx bar/ θ has x^2 (chi) distibution with 2n degrees of freedom.
d) based on part (c),find a 95 % confidence interval for θ if n=10
Please help with the attached question. Don’t have to rewrite the question, it’s just fill the blanks
Let X,, = U {k —;1;,k+ i) for any n E N. Prove that I] X,1 = Z by filling out the gaps in the#152 116"argument below. Proof. For any n E N and 1′ E R we have _< :1: <_. In particular, if :1: E Z, then :r E __ forany n E N as _. Therefore, E C X,1 for any n E N. As a result, _. We want to show that n X“ C Z. From the equivalence of the “if-then” statement and itsnEN contrapositive, it is enough to show that _. Notice that n Xn C IR as for all k E Z and allHEN n E N we have _. Therefore, we want to show that if 1: E_, then _. Assume r E_. Then,choose k E E such that k < r < k + 1. In particular, a.“ — k_ and (k +1)— :t_, so we canchoose or E N such that _ and _ because _. As a result, a >_ and a: <_. Therefore, 1: 5-3 _. So, :2: ¢_. Since _ and _, we have _. III
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