1

Suppose I have $X_1\sim Uniform[0,1]$.
Let us define a sequence of random variables as follows:

We will choose a random point from the interval $[0,X_1]$ and mark it by $X_2$. then we will choose a random point from the interval $[0,X_2]$ and mark it by $X_3$ and so on. 0

I need to find a formula for the density function $f_{{X}_n}$.

I could see that $X_n$ has more probability to be near 0.
I tried to express $F_{X_n}$ in terms of $F_{X_{n-1}}$ and $F_{X_{n-2}}$ but got only to this equation: $F_{X_{n}}(x_{n+1})=\frac{F_{X_{n+1}}(x_{n+2})}{F_{X_{n-1}}(x_n)}$ that leads me nowhere even if I trying to derivate it.

Thank you,
Michael

michael
  • 211
  • This question was asked on this site a couple of days ago. Unfortunately I can't find the duplicate. Please let us know how you came upon this question. – joriki Jun 07 '18 at 06:16
  • I tried to search for a similar question and didn't find anything... By the way I didn't understand why you asking me how I came upon this question – michael Jun 07 '18 at 06:21
  • See here : https://math.stackexchange.com/questions/2810381/pdf-of-a-member-of-a-sequence-of-dependent-random-variables – nicomezi Jun 07 '18 at 06:32
  • @michael: I asked because when the same question turns up on the site more than once within days, that's often because it's a question in an ongoing contest, an exam or the like. That's not necessarily due to ill intention by the poster; some people just don't know about our contest problem policy. But quite apart from that, it's always good to provide context for your question. I wasn't implying that you'd necessarily done something wrong. – joriki Jun 07 '18 at 08:21
  • @joriki ahh...I didn't know about this policy. This question was asked by our lecturer in probability theory course – michael Jun 07 '18 at 09:51

0 Answers0