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Please can someone expalin how i do question 8i in this picture, I have tried finding X in terms of theta but don't know then how to turn this into a pdf (if that is even what it is looking for)? I have basically got to x=δtan(θ)but don't know how to contiune?? Please help?

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    To sum up, the question is to compute the PDF of x=δ.tan(θ) when θ is uniform on $(0,\pi/2)$ (or on $(-\pi/2,\pi/2)$, this is not quite clear). Which approaches do you know to do that? – Did Apr 09 '14 at 10:31

2 Answers2

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Hint: Given $\delta$, If we assume that $\theta$ is uniformly distributed over $[\frac{-\pi}2 \frac{\pi}2]$, the point $x$ is distributed according to Cauchy distribution: $$ f(x) = \frac{1}{\pi\delta \left[1 + \left(\frac{x}\delta\right)^2 \right]}. $$

Arash
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Hint:

$$\delta\tan\theta\leq x\iff\theta\leq\arctan\frac{x}{\delta}$$

That gives you a possibility to find the cdf. For pdf you take its derivative.

drhab
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