The joint distribution for the lenght of life of two different types of componen

Question

The joint distribution for the lenght of life of two different types of components operating in a system is given by                       0                              , elsewhere
The relative efficiency of the two types of components is measured by U= Y2 / Y1.
a.) Let V = Y1. Find the joint density for U and V. b.) Find the marginal density for U. c.) Find E(V|U = 1) The joint distribution for the lenght of life of two different types of components operating in a system is given by f(y1, y2) = {1/8 y1 e -(y1 + y2)/2 The relative efficiency of the two types of components is measured by U = Y2 / Y1. Let V = Y1. Find the joint density for U and V. Find the marginal density for U. Find E(V|U = 1)

Explanation / Answer

:a) Let U = Y?/Y? and V = Y? <==> Y? = V and Y? = UV.

So, the boundary lines transform as follows:

y? = 0 ==> v = 0

y? = 0 ==> u = 0.

(Moreover any point in the first quadrant of (y?, y?) is mapped to a point in the first quadrant of (u, v).)

Next, the Jacobian ?(y?, y?)/?(u, v) equals

|0 1|

|v u| = -v.

So, g(u, v) = f(y?, y?) * |?(y?, y?)/?(u, v)|

................= (1/8)y? e^(-(y? + y?)/2) * |-v|

................= (1/8)v e^(-(v + uv)/2) * v

................= (1/8)v^2 e^(-v(1+u)/2) for u, v ? 0 (and 0 otherwise)

--------------

b) Integrating through the domain of V,

g_u(u) = ?(v = 0 to ?) (1/8)v^2 e^(-v(1+u)/2) dv

..........= ?(t = 0 to ?) (1/8) (2t/(1+u))^2 e^(-t) * 2 dt/(1+u), letting t = v(1+u)/2

..........= (1+u)^(-3) * ?(t = 0 to ?) t^2 e^(-t) dt

..........= (1+u)^(-3) * 2, via gamma function or repeated integration by parts

..........= 2/(1+u)^3 for u ? 0 (and 0 otherwise).

--------------

c) g_v(v | U = 1) = [(1/8)v^2 e^(-v(1+u)/2)] / [2/(1+u)^3] {at u = 1}

........................= (1/2)v^2 e^(-v) for v ? 0 (and 0 otherwise).

So, E[V | U = 1]

= ?(v = 0 to ?) v * (1/2)v^2 e^(-v) dv

= (1/2) * ?(v = 0 to ?) v^3 e^(-v) dv

= (1/2) * 3!

= 3.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.