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1. Density function of random variable transformation

Given $X$ a real valued random variable with a density function $f_X$. Let $g$ be a continuous function.

Does this imply, that the random variable $g(X)$ has a density function? Or do I need more assumptions, for example that $g$ has an inverse function?

2. Density function of marginal distribution

Let $\lambda(dx,dy)$ be a Radon measure with a density function and $D$ is compact. Does this imply, that the marginal distribution $\lambda(dx)=\int_D \lambda(dx,dy)dy$ has a density function too?

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If $g$ is a cosntant then $g(X)$ does not have a density. Even if you assume that $g$ has an inverse $g(X)$ need not have a density. For example if $X$ has uniform distribution and $g$ is a striclty increasing singular function then $g(X)$ does not have a density.

For the second part the answer is YES. $m(A)=0$ implies $m_2(A\times \mathbb R)=0$ where $m_2$ is two dimensional Lebesgue measure. This implies $\lambda (A \times \mathbb R)=0$ so the first marginal of $\lambda$ has an absolutely continuous distribution.

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  • $\begingroup$ Thank you, that helps! $\endgroup$ – freeba Aug 12 at 13:47

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