Problem 1 Let W (t) be the standard Brownian Motion. Note that W (t) is a Wiener process. (a) Find P(-4 < W(3) < 4) (b) Find P(3 < W (3) + W (7) < 7) (c) Find P(W (3) > 3 | W(4) = 2)

Q: Find the critical points of ƒ(x,y) = (x³y-y³) 5

A: To determine the critical points.

Q: If F(x) = f(g(x)), f(2) = 8, f'(2) = 4, g(5) = 2 and g' (5) = –2, then F' (5) is %3D %3D 16 -8 12

A:

Q: Show that the Stein operator for Poisson P(Y = k) = is Tf(r) = Af(k + 1) – kf(k).

A: Given information: The Stein operator for Poisson P(Y = k) = λk/k! is Tµf(x) = λf(k + 1) – kf(k).

Q: g(y) y - 1 y²y + 1 2

A: to find critical points of gy=y-1y2-y+1

Q: Assume that f(x) and g(x) are differentiable for all x. 2f(x) g(x) + 1 Let h(x) = find h' (x).

A:

Q: 1 7. Let f(x) = and g(x) 1-x 1 Show that if b + 0, %3D %3D ax +b then g(x) = f (-;x).

A: f(x)=11-x , g(x)=1ax+b

Q: f(x,v) = x² sin(x + y²) Po= (xo, Yo) = (2, 1) n=0.05 The gradient descent algorithm can be written…

A: Given ,f(x, y ) = x2 sin (x+y2)P0 = (x0 , y0) = (2 , 1)η = 0.05gradient descent algorithmPn =…

Q: Question 3 A set of n functions f (x), ƒ₂(x), ..., f (x) is. O linearly dependent. O linearly…

A: The given definition is of linearly dependent or linearly independent.

Q: 1 Integrate with respect to x. 4+ 25(3x +1)²

A:

Q: Provide an interpretation for the following SPSS Output: Total Annual Household Income (in USD) Age…

A: Part 2: Explanation:Step 1: Understanding Pearson Correlation CoefficientThe Pearson correlation…

Q: (2)Z/(5)=........... Ο (5) 5Z Ο {[0],[1],[2],[3],[4]} O Ο Ο *

A: To find: Z/(5)

Q: 5. Find the moment generating function of P,(x) = p(1– p)*-* ko.) (x) where k 2 l %3D

A: Moment generating function :

Q: find critical points- q(t) = (t2+4)/(t2-4)

A: We have to find

Q: 6) Continuing, we let

A: Chain rule for total derivative:  Assume that F:Rm→Rn and G:Rn→Ro are maps, with F differentiable at…

Q: x(t) = Find real-valued closed formulas for the trajectory (t + 1) = Ax(t), where z (0) = [!] A 9 12…

A:

Q: Let f(x) = x² - 3, g(x) = Vx+1, h(x) : x+2 3 j(x) = -3x-5 %3! Determine the followings: ) f(g(3)) b)…

A: Here we want to find the composition of the given function.

Q: If y is a solution of y(x)-[(x-1t)»(?)dt = 1, then which of the following is true? (a) y is bounded…

A:

Q: Is ƒ continuous at x = -1?

A: From the graph, we see that x=-1 is an endpoint  And it has a closed circle. That means the function…

Q: Q 6. If the function f(x) = 2x - 9ax + 12a*x+1 has a local maximum at x x1 and a local minimum at x=…

A:

Q: (b) Let f:(R\{1}) → (R\{1}) and ƒ˜¹:(R\{1} ) → (R \{1}) be both defined by f(x) = f¯¹(x) = = x+1…

A: As both function and its inverse are given in the question so we have to determine that…

Q: 4) Consider the function F(x) = f f (t) dt . Find F¹ (7). O F(7)-F(2) O f(x) O f(7) Of(7)-f(2)

A: Given query is to find indicated derivative.

Q: Suppose that X ~ fx(x) = 3x²1(0 < x < 1). (a) Verify that fx is a PDF. (b) Derive and sketch the CDF…

A:

Q: 1 1 Let f (z) = and I = P.V. S dx. (x2+1)3(x2-3x+2)x (z2+1)3(z2-3z+2)z If p = Res f(z)|z=-i» r = Res…

A:

Q: Describe the long run behavior of f(x) = 2(4)ª + 4: As x →→∞, f(x) → As x → ∞, ƒ(x) → 0 ✓ X -00

A: