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sanctifier
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Homework Statement
The probability desity function (p.d.f.) of joint distribution of random variables X and Y is given as
[itex] f(x,y) = \begin{cases} e^{-(x + y)} \;\; when \;\; x > 0 \\ 0 \;\; \;\;\;\;\;\;\;\;\;\;otherwise \end{cases} [/itex]
Question 1: What are the p.d.f.'s of X + Y and X/Y ?
Question 2: Does the expectation of X/Y exist ?
Homework Equations
Nothing special.
The Attempt at a Solution
Answer 1:
[itex] \begin{cases} u = x \\ v = x + y \end{cases} [/itex]
[itex] \begin{cases} x = u \\ y = v - u \end{cases} [/itex]
[itex] Jacobian = \begin{bmatrix} \frac{dx}{du} & \frac{dx}{dv} \\\frac{dy}{du} & \frac{dy}{dv} \end{bmatrix} = \begin{bmatrix}1 & 0 \\{-1} & 1 \end{bmatrix} [/itex]
[itex] g(u,v)=f(u,v-u)|Jacobian|= e^{-v} [/itex]
[itex] h(x+y)=h(v) = \int_0^v g(u,v)du = e^{-v} u |_{u=0}^{u=v} = ve^{-v} [/itex]
[itex]\begin{cases}z = x\\ w = \frac{x}{y} \end{cases} [/itex]
[itex]\begin{cases} x = z\\ y = \frac{z}{w} \end{cases} [/itex]
[itex]Jacobian2 = \begin{bmatrix} \frac{dx}{dz} & \frac{dx}{dw} \\\frac{dy}{dz} & \frac{dy}{dw} \end{bmatrix} = \begin{bmatrix}1 & 0 \\{ \frac{1}{w} } & {- \frac{z}{ w^{2}} } \end{bmatrix}[/itex]
[itex]g2(z,w)=f(z, \frac{z}{w} )|Jacobian2|= e^{-z- \frac{z}{w} } \frac{z}{ w^{2} } [/itex]
[itex]h2(w)= \int_0^ \infty g2(z,w)dz = \int_0^ \infty e^{-z- \frac{z}{w} } \frac{z}{ w^{2} } dz = -\int_0^ \infty \frac{z}{ w^{2} } {(1+ \frac{1}{w} )}^{-1} d e^{-z(1+ \frac{1}{w})} = 0 + \int_0^\infty \frac{e^{-z(1+ \frac{1}{w})}}{ w^{2} \frac{1}{w}} dz = - \frac{1}{ {w+1}^{2} } e^{-z(1+ \frac{1}{w})} |_{z=0}^{z= \infty }= \frac{1}{{w+1}^{2}} [/itex]
Answer 2:
[itex]E( \frac{x}{y} )=E(w)= \int_0^\infty \frac{1}{{w+1}^{2}} dw = -\frac{1}{w+1}|_{w=0}^{w=\infty} = 1[/itex]
Are the two answers correct? Thank you in advance.