5.9 We define g(z, v) := log p(æ, z) – log q(z, v) z := t(e, v) for differentiable functions p, q, t, and æ € RP, z € RE, ve RF, € € RG. By using the chain rule, compute the gradient d ,9(z, ν) . dv

Q: (a) express dw/ dt as a function of t, both by using the Chain Rule and by expressing w in terms of…

A: Given- w=x2+y2, x=cost, y=sint and t=π To Find- The value of dwdt as a function of t, both by using…

Q: Suppose that T(0) = a and T(1) = b are some constants. Define the running pairwise average as, for n…

A: Given: Tn+2=12Tn+1+Tn for n≥0 T0=a T1=b To find: The limit of Tn as n→∞.

Q: In each of the following find the pdf of Y. (a) Y = X? and fx(x)= 1, 0 < I< 1 (b) Y = - log X and X…

A:

Q: Let X1,..., Xn be iid and U([a, b])-distributed with unknown parameters 01 = a and 02 = b. (a) Use…

A: (a). Use the method of moments to derive T(1) and T(2) for θ1 and θ2: Given that X1, X2, … , Xn be…

Q: af Q1: a) Use the chain rule to calculate for the function f(x, y, z) = 3x² - 3y + 4xz², x =…

A:

Q: R SECTION 2. 1 1. Compute the gradients of the following functions at the given points. (a) f(x,y)=…

A: Let ft be a C1 function such that f't≠0 Given Fx,y=fx2+y2                      (i) It is known that…

Q: For the given functions, (a) express dt as a function of t, both by using the chain rule and by…

A: Topic = Derivative

Q: 4. Suppose you hang a bucket from a spring that stretches it 15 mm from its natural length. You then…

A: D=nW+15 Weight (N) 10 15 20 25 30 35 Distance (mm) 57.4 67.6 71.6 104.5 106.9 136.0 Plot the…

Q: Q2: Discuss the fundamental theorem for gradients using E = 5x2 i + 20 xy j+ 10 yz3 k and the…

A: Fundamental theorem of gradient states that a if  line integral through a  F gradient field depends…

Q: 17. Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further…

A:

Q: 3) A hot cup of tea has temperature T(1) = 198 °F, one minute after it has been sitting on the…

A: To calculate the local linearization of the temperature at t = 1.

Q: 3. Use definition (not tables) of L{f(t)}= [ e"f (t)dt to find L{1*e" } . Assume s>9

A:

Q: 4. Recall that (In z) = 1, 4(e)=e, and that (a") = a* In a, where a>0 is a constant (as deduced in…

A: From the given problem:   ddxcos6x   Now applying the chain rule:…

Q: 6.5.4 Find the PDF of W = X + Y when X and Y have the joint PDF fx,x(z,y) • otherwise.

A: Given: The joint probability density function of the random variables X and Y is provided as:…

Q: We have found the following Wronskians. W = -e-³x; W₁ =-e-2x(4+ex)-¹; W₂ = e¯x(4+ e*)-¹ We want to…

A:

Q: 1. Show that va(z(t), t) = z(t)P"(t)Q(t)P(t)z(t) is a Lyapunov function that proves uniform…

A: Sol

Q: If T, is an MVUE of y (0) and T, is any other unbiased estimator of y (0) with efficiency e < 1,…

A: Answer: For the given data,

Q: Find the Wronskian for the set of functions {e¬*, xe¯×, (x + 1)e-*}.

A:

Q: The residual of ŷ; is defined by e, = y, – ĝ.. (1) Prove Σ -0 and Σiι u ΣΗ. (2) Prove E1 T;ei = 0…

A: (1) In problem 1 we have to prove that : a) The sum of residuals in regression model that contain an…

Q: Show that the Cobb-Douglas production function z = Cxay1−a can be rewritten as        ln z/y = ln C…

A: Given: z=cxay1-a for converting given function in log we simplify it and then take log both side so,…

Q: 2. If A = f+ g,+ hy s.t f(x.y.z) = e2x, g(x.y.z) = xcosy, h(x. y. z) = log 2z then div A %3D %3D

A:

Q: (a) express dw/ dt as a function of t, both by usingthe Chain Rule and by expressing w in terms of t…

A: A) The function x is differentiated as, x=cost⇒dxdt=-sint The function y is differentiated as,…

Q: 6. Find the Wronskian for the set of functions {e¬x, xe¯*, (x + 1)e¬*}.

A:

Q: In each of the following, find pdf of Y a) Y= X2 and  fX(x) = 1, 0 0 c) Y = eX  and fX(x) = 1/σ2 *…

A: Given: Then:

Q: 1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

A:

Q: 1) Create a in solving ODE 3 of the for m: flow chart / road map y" + t p(a) y' (2) = 0 .

A:

Q: . In one model of the changing population P(t) of a community, it is assumed that 4 = -, where and…

A: As we have to solve for p(t) in (a) so there is clearly  a misprint it should be dD/dt in place of…

Q: 2. Define the p.m.f and give the values of u , o ? , and o when the moment generating function of X…

A: Given MGF is, M(t)=25+35et The given MGF is same as the MGF of the Bernoulli distribution,…

Q: Suppose z = x² sin y, x = 3s² +lt², y= 2st. A. Use the chain rule to find and as functions of x, y,…

A: Consider:z=x2sin y, x=3s2+t2, y=2st. A1) Find ∂z∂s. Apply chain rule ∂z∂s=∂z∂x·∂x∂s+ ∂z∂y·∂y∂s Find…

Q: (a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A: We find dw/dt using chain rule. Here plug x = cos t, y = sin t, z = 4sqrt(t) and simplify.

Q: Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in…

A: You have correctly answered the first part. I will straightway go to the second part.

Q: Suppose that the constant rate of change of y with respect to z is 4.6. Based on the above…

A: Rate of change of any function is the first derivative of the function with respect to variable on…

Q: Suppose there is no known formula for g(x), but we know that g(2) = -1 and d (z) = V2 +5 for all x.…

A:

Q: Compute the following values for the given function. h(s, t) = 2s ln(t) − 7t ln(s) h(1, e) =…

A: a) Given that s=1 and t=e  Then we plug the values in the formula and simplify

Q: Suppose solving an equation by Laplace transform results in 4 (s+ 18) Y (s) : %3D (s+9)² * Evaluate…

A: Laplace transform of the function y(t) is defined as Y(s)=∫0∞e-sty(t)dt, where s&gt;0. Since the…

Q: 4. In one model of the changing population P(t) of a community, it is assumed -, where and are the…

A:

Q: The function that models the amount of money C that is obtained by investing P money at an annual…

A: We have to find the minimum investment.

Q: 2. Consider the model: Y, =0.2Y+e,+0.4e. Find the autocorrelation function p, for k=1,2,3.

A: We are required to find autocorrelation function(acf) of an ARMA model.

Q: For a given function P = f(t), it is known that 0 dt2 dt Sketch a possible graph of the function P…

A: Known fact: The graph of the function y=fx is decreasing on the interval I if dydx&lt;0 for all x in…

Q: Let X1,..., Xn be iid and U([a, b])-distributed with unknown parameters 01 (a) Use the method of…

A:

Q: Consider the following. w = In(x2 + y), x = 2t, y = 8 - t dw by using the appropriate Chain Rule. dt…

A: Solution is given below..

Q: Suppose that T(0) = a and T(1) = b and are some constants. Define the running pairwise average as,…

A: The function of T is determined using the given recurrence relation as shown below.

Q: 5.9 We define g(z,v) := log p(æ, z) – log q(z,v) z:= t(e, v) for differentiable functions p, q, t.…

A:

Q: 1.4 Using the graph in Figure 1.7 below, find SLaf (t)dt and if the area of the shaded region is A,…

A:

Q: For the AR(2) process Y, = (1/3)Y–1+(2/9)Y-2+ Et, find the autocorrelation function of Y; and…

A: Given the AR(2) process as Yt=13Yt-1+29Yt-2+εt

Q: Let g(x,y)=ye-3x, x(t) =ln(t2+1), and y(t) =√t Find g(x(t), y(t)

A: Given,x(t)=lnt2+1y(t)=tg(x,y)=ye-3x

Q: Prove that by Gamma Function eit + e' e-st dt 2 s2 + 1

A: We will find out the required result using Gramma function.

Q: Prove that by Gamma Function pit +e e-st dt %3D s2 +1

A: Gamma Function: The gamma function, denoted by is denoted by which is convergent for Some…