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I have the following function: $$h(t)=L(t)\dfrac{EFL}{d_0-v(t)t}$$ $EFL$ and $d_0$ are know constants. $L(t)=L_0+\nu(t)$ and $v(t)=v0+w(t)$ where $L_0$ and $v_0$ are constant. $\nu(t)$ and $w(t)$ are random variables normally distribuited. So: $$\nu(t)=\mathcal{N(0,\sigma_0)}$$ and $$w(t)=\mathcal{N(0,\sigma_1)}.$$ What is the PDF of $h(t)$? Thanks.

Riccardo.Alestra
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1 Answers1

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One has to consider $h(t)$ as a function of two independent random variables $$ g(t)=\nu(t) $$ and $$ h(t)=(L_0+\nu(t))\frac{EFL}{d_0-v_0t-w(t)t}. $$ Then, the Jacobian is given by $$ J=\frac{t}{EFL}\frac{h^2}{L_0+g}. $$ Then, the distribution becomes $$ P(h,g)=\frac{t}{EFL}\frac{h^2}{L_0+g}\frac{1}{\sigma_0\sqrt{2\pi}}e^{-\frac{g^2}{2\sigma_0^2}}\frac{1}{\sigma_1\sqrt{2\pi}}e^{-\frac{1}{2\sigma_1^2t^2}\left(d_0-v_0t-EFL\frac{L_0+g}{h}\right)^2}. $$ Your distribution will be given by $$ H(h)=\int_{-\infty}^\infty P(h,g)dg. $$

Jon
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