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8. Let U be uniformly distributed on [0, L], where L has Exponential(A) distribution. Let V- L - U. What is the joint density function of U and V?
8. Let U be uniformly distributed on [0, L], where L has Exponential(A) distribution. Let V- L - U. What is the joint density function of U and V?
In the given circuit, identify (0) and i(t) for t> 0. Assume 10) = 0 V and 1(O) = 2.50 A. + 5u(t) A 222 v+0.5 F ell 1 H [3.780 e-t2cos(1.3229t - 90°)]u(1) V [5 + 2.67252 e-t2cos(1.3229t - 200.79] A [0 – 2.67252e-t2cos(1.3229t – 200.7°)] A [5 – 2.67252e-t/2 cos(1.3229t+ 200.79] A [2.673e-t2cos(1.3229t – 90°)]u(0) V [1.336e-t2 cos(1.3229t+ 90°)]/(t) v
Pivotal quantity for a double exponential distribution: 5 points Assume y follows a double exponential distribution (as above), where is the parameter of interest and is unknown, and o is known to be 1. Show that Y -u is a pivotal quantity.
Find the laplace transform of the following
function
I(t) = 10^4 t *u(t)-10*4 t + 100*10^-6 *
u(t-10*10^-9)
Just to clarify that is a negative 6 in the
exponential term
이이 (0 | |
(1 point) Solve the heat problem with non-homogeneous boundary
conditions
∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0
u(0,t)=0, u(3,t)=2, t>0,u(0,t)=0, u(3,t)=2, t>0,
u(x,0)=23x, 0<x<3.u(x,0)=23x, 0<x<3.
Recall that we find h(x)h(x), set
v(x,t)=u(x,t)−h(x)v(x,t)=u(x,t)−h(x), solve a heat problem for
v(x,t)v(x,t) and write u(x,t)=v(x,t)+h(x)u(x,t)=v(x,t)+h(x).
Find h(x)h(x)
h(x)=h(x)=
The solution u(x,t)u(x,t) can be written as
u(x,t)=h(x)+v(x,t),u(x,t)=h(x)+v(x,t),
where
v(x,t)=∑n=1∞aneλntϕn(x)v(x,t)=∑n=1∞aneλntϕn(x)
v(x,t)=∑n=1∞v(x,t)=∑n=1∞
Finally, find
limt→∞u(x,t)=limt→∞u(x,t)=
Please show all work.
(1 point) Solve the heat problem with non-homogeneous boundary conditions au ди (x, t) at (2, t), 0<x<3, t> 0 ar2 u(0,t) = 0, u(3, t) = 2, t>0, u(t,0)...
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
Calculate i(t) for t> 0 in the given circuit. Assume A = 35[1 – u(t)] V. + V - 1 F 16 A +1 H 592 The value of i(t) = (A) cos (Ct + Dº)u(t) A where A = C= and D =
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...