let's say I have U~Uniform(0,1) and W~Uniform(0,U), I have a few questions about it

  • How can I determine the conditional distribution of W given U? i.e. FW|U(w|u) = P[W ≤ w|U = U]
  • How can I find the density function of W given U
  • Finding the unconditional density function of W

For the first part, by bayes:

$$P(U > s | U > t) = \frac{P\left[\{U > s\}\cap \{U > t\} \right]}{P(U > t)}$$

so it follows

$$P\left[\{U > s\}\cap \{U > t\} \right]=P(U > s)$$

But I am not sure how to continue or finish if this is correct

For the second question, I assume I have to use the formula for a conditional distribution, however I think this is wrong as well


For the last questions, I am unsure about how to answer it at all.

Any help is appreciate, thank you!


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