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I defined a PDF called dist, for example like this one:

dist = ProbabilityDistribution[
  Convolve[PDF[NormalDistribution[4, 5], x], 
   PDF[NormalDistribution[3, 1], x], x, t], {t, -Infinity, Infinity}]

Is it possible to get also a shifted PDF? For example

$p_1(x)=dist(x), p_2(x)=dist(x-c)$

where c is a constant

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A transformed distribution should do what you want, e.g.,

dist=TransformedDistribution[a u + b,  u \[Distributed] NormalDistribution[mu, sigma]]

so in your case, distributed by your custom PDF. Be aware, while neat, MMA fancy-probability features have their limits...

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