let's play a (continuous) probability game!












2












$begingroup$


Here's the description of the game.



We have a target number $x$ in mind and start by picking a number $c_1$ uniformly at random from $(0, 1)$. Then $c_2$ is picked uniformly at random from $(0, c_1)$, and in general, $c_i$ is picked uniformly at random from $(0, c_{i -1})$. The game stops when we pick a $c_i$ such that $c_i < x$.



What is the expected number of times we'll have to pick a number $c_i$?
Alternatively, what is the expected length of the sequence of $c_i$'s?



I am guessing that the solution will be in terms of our target $x$. How would one go about approaching this problem? I thought about conditioning in terms of $c_1$, but I couldn't work it out.



Edit: we have $x in (0, 1)$.



Edit 2: I am not hell-bent on using a conditional expectation approach. But if there exists some ridiculously simple solution, I am guessing it would involve it.










share|cite|improve this question











$endgroup$












  • $begingroup$
    How far have you gotten with this?
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:34










  • $begingroup$
    @saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
    $endgroup$
    – 0k33
    Dec 2 '18 at 16:35










  • $begingroup$
    Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:42






  • 1




    $begingroup$
    @0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
    $endgroup$
    – Théophile
    Dec 2 '18 at 20:07








  • 1




    $begingroup$
    @Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
    $endgroup$
    – 0k33
    Dec 2 '18 at 20:18
















2












$begingroup$


Here's the description of the game.



We have a target number $x$ in mind and start by picking a number $c_1$ uniformly at random from $(0, 1)$. Then $c_2$ is picked uniformly at random from $(0, c_1)$, and in general, $c_i$ is picked uniformly at random from $(0, c_{i -1})$. The game stops when we pick a $c_i$ such that $c_i < x$.



What is the expected number of times we'll have to pick a number $c_i$?
Alternatively, what is the expected length of the sequence of $c_i$'s?



I am guessing that the solution will be in terms of our target $x$. How would one go about approaching this problem? I thought about conditioning in terms of $c_1$, but I couldn't work it out.



Edit: we have $x in (0, 1)$.



Edit 2: I am not hell-bent on using a conditional expectation approach. But if there exists some ridiculously simple solution, I am guessing it would involve it.










share|cite|improve this question











$endgroup$












  • $begingroup$
    How far have you gotten with this?
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:34










  • $begingroup$
    @saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
    $endgroup$
    – 0k33
    Dec 2 '18 at 16:35










  • $begingroup$
    Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:42






  • 1




    $begingroup$
    @0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
    $endgroup$
    – Théophile
    Dec 2 '18 at 20:07








  • 1




    $begingroup$
    @Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
    $endgroup$
    – 0k33
    Dec 2 '18 at 20:18














2












2








2


2



$begingroup$


Here's the description of the game.



We have a target number $x$ in mind and start by picking a number $c_1$ uniformly at random from $(0, 1)$. Then $c_2$ is picked uniformly at random from $(0, c_1)$, and in general, $c_i$ is picked uniformly at random from $(0, c_{i -1})$. The game stops when we pick a $c_i$ such that $c_i < x$.



What is the expected number of times we'll have to pick a number $c_i$?
Alternatively, what is the expected length of the sequence of $c_i$'s?



I am guessing that the solution will be in terms of our target $x$. How would one go about approaching this problem? I thought about conditioning in terms of $c_1$, but I couldn't work it out.



Edit: we have $x in (0, 1)$.



Edit 2: I am not hell-bent on using a conditional expectation approach. But if there exists some ridiculously simple solution, I am guessing it would involve it.










share|cite|improve this question











$endgroup$




Here's the description of the game.



We have a target number $x$ in mind and start by picking a number $c_1$ uniformly at random from $(0, 1)$. Then $c_2$ is picked uniformly at random from $(0, c_1)$, and in general, $c_i$ is picked uniformly at random from $(0, c_{i -1})$. The game stops when we pick a $c_i$ such that $c_i < x$.



What is the expected number of times we'll have to pick a number $c_i$?
Alternatively, what is the expected length of the sequence of $c_i$'s?



I am guessing that the solution will be in terms of our target $x$. How would one go about approaching this problem? I thought about conditioning in terms of $c_1$, but I couldn't work it out.



Edit: we have $x in (0, 1)$.



Edit 2: I am not hell-bent on using a conditional expectation approach. But if there exists some ridiculously simple solution, I am guessing it would involve it.







probability probability-theory statistics probability-distributions conditional-probability






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 2 '18 at 20:45







0k33

















asked Dec 2 '18 at 16:19









0k330k33

12010




12010












  • $begingroup$
    How far have you gotten with this?
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:34










  • $begingroup$
    @saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
    $endgroup$
    – 0k33
    Dec 2 '18 at 16:35










  • $begingroup$
    Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:42






  • 1




    $begingroup$
    @0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
    $endgroup$
    – Théophile
    Dec 2 '18 at 20:07








  • 1




    $begingroup$
    @Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
    $endgroup$
    – 0k33
    Dec 2 '18 at 20:18


















  • $begingroup$
    How far have you gotten with this?
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:34










  • $begingroup$
    @saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
    $endgroup$
    – 0k33
    Dec 2 '18 at 16:35










  • $begingroup$
    Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
    $endgroup$
    – saulspatz
    Dec 2 '18 at 16:42






  • 1




    $begingroup$
    @0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
    $endgroup$
    – Théophile
    Dec 2 '18 at 20:07








  • 1




    $begingroup$
    @Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
    $endgroup$
    – 0k33
    Dec 2 '18 at 20:18
















$begingroup$
How far have you gotten with this?
$endgroup$
– saulspatz
Dec 2 '18 at 16:34




$begingroup$
How far have you gotten with this?
$endgroup$
– saulspatz
Dec 2 '18 at 16:34












$begingroup$
@saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
$endgroup$
– 0k33
Dec 2 '18 at 16:35




$begingroup$
@saulspatz the only idea I have is trying to apply some form of conditional expectation based on the first number picked (I've seen a somewhat similar, but much easier, problem before that used that approach). I couldn't flesh it out much less provide an equation for what that would look like. The question is very unintuitive to me.
$endgroup$
– 0k33
Dec 2 '18 at 16:35












$begingroup$
Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
$endgroup$
– saulspatz
Dec 2 '18 at 16:42




$begingroup$
Same here. It seems like it shouldn't be hard, but I can't get a handle on it. somehow.
$endgroup$
– saulspatz
Dec 2 '18 at 16:42




1




1




$begingroup$
@0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
$endgroup$
– Théophile
Dec 2 '18 at 20:07






$begingroup$
@0k33 Yes, certainly the derivation is the interesting part (I've added mine below)... I wonder if there is an easier way though?
$endgroup$
– Théophile
Dec 2 '18 at 20:07






1




1




$begingroup$
@Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
$endgroup$
– 0k33
Dec 2 '18 at 20:18




$begingroup$
@Théophile thank you so much for your explanation! I was able to follow all of it. At the risk of sounding like a broken record, I do think the easier solution would have to do with conditional expectation (in fact I was suggested to use that approach). I have been beating my head on this problem all morning and cannot for the life of me see how, though.
$endgroup$
– 0k33
Dec 2 '18 at 20:18










2 Answers
2






active

oldest

votes


















2












$begingroup$

Given $x$, let $f(x)$ be the expected length of the game. We'll show that $f(x) = 1 - ln x$.



The probability that we finish the game on the first step is $x$; the game continues with probability $1 - x$. We therefore have
$$f(x) = 1 + (1 - x)cdot(textrm{#future moves}).$$



Now, suppose we have to continue, i.e., $c_i in (x, 1)$. Rather than choosing the next number $c_{i+1}$ from $(0, c_i)$, we can instead scale $x$ appropriately so that $c_{i+1}$ is chosen from $(0, 1)$, effectively restarting the game with $x/c_i$ instead of $x$.



To account for the infinitely many ways to choose $c_i in (x, 1)$, we can take the following integral:
$$textrm{#future moves} = frac1{1-x}int_x^1fleft(frac xuright) du$$



In other words,
$$f(x) = 1 + int_x^1fleft(frac xuright) du.tag{1}$$



The rest (I'm skipping a few steps here...) is a matter of manipulating this to eventually bring it into the form of a differential equation,
$$xf''(x) = -f'(x),$$
whose general solution is $f(x) = a + bln x$. From there, using $(1)$ we find $a=1, b=-1$, solving the problem.



Given the simplicity of the final expression there may be a much more direct solution, but at the moment I don't see one.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    This answer uses the following statements:




    Lemma 1: Let $U_1,ldots,U_n$ be independent and identically distributed uniform random variables on the interval $(0,1)$. Then the probability density function of the product $U_1 ldots U_n$ is given by $$ f_n(z) = frac{(-1)^{n-1}}{(n-1)!} log^{n-1}(z) 1_{(0,1)}(z).$$



    Lemma 2: $$int log^{n}(z) , dz = z sum_{k=0}^n (-1)^{n-k} frac{n!}{k!} (log z)^k, qquad n in mathbb{N}$$




    Hints: Let $(U_i)_{i in mathbb{N}}$ be a sequence of independent random variables which are uniformly distributed on $(0,1)$.




    1. Define iteratively $$C_1 := U_1 qquad C_i := U_i C_{i-1}, qquad i geq 2.$$ Check that $(C_i)_{i in mathbb{N}}$ can be used to model the outcome of your probability. Show that $$C_i = prod_{j=1}^i U_j$$ for any $i in mathbb{N}$.

    2. Define $$tau := inf{i in mathbb{N}; C_i < x}.$$ Use the fact that the sequence $C_i$ is strictly increasing in $i$ to show that $$mathbb{P}(tau geq i+1) = mathbb{P}(C_i geq x).$$

    3. By Step 1 and Lemma 1, we have $$mathbb{P}(C_i geq x) = frac{(-1)^{i-1}}{(i-1)!} int_x^1 log^{i-1}(z) , dz$$ Use Lemma 2 to conclude that $$mathbb{P}(C_i geq x) = 1- x sum_{k=0}^{i-1} (-1)^k frac{1}{k!} (log x)^k quad text{for $i geq 2$}.$$


    4. Since $$mathbb{P}(tau=i) = mathbb{P}(tau geq i)-mathbb{P}(tau geq i+1)$$ it follows from Step 2 and 3 that $$mathbb{P}(tau=i) = x (-1)^{i-1} frac{(log x)^{i-1}}{(i-1)!} quad text{for $i geq 3$}.$$ Show that $$mathbb{P}(tau=1) = x qquad mathbb{P}(tau=2) = -x log x$$


    5. Deduce from Step 4 that $$mathbb{E}(tau) = sum_{i in mathbb{N}} i mathbb{P}(tau=i) = x +x sum_{i geq 2} i (-1)^{i-1} frac{log(x)^{i-1}}{(i-1)!}$$


    6. Using $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = frac{d}{dz} sum_{i geq 2} frac{z^i}{(i-1)!}$$ prove that $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = exp(z) (z+1)-1 $$


    7. Conclude that $$mathbb{E}(tau) = 1- log(x).$$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
      $endgroup$
      – 0k33
      Dec 3 '18 at 18:44






    • 1




      $begingroup$
      @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
      $endgroup$
      – saz
      Dec 3 '18 at 19:33











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Given $x$, let $f(x)$ be the expected length of the game. We'll show that $f(x) = 1 - ln x$.



    The probability that we finish the game on the first step is $x$; the game continues with probability $1 - x$. We therefore have
    $$f(x) = 1 + (1 - x)cdot(textrm{#future moves}).$$



    Now, suppose we have to continue, i.e., $c_i in (x, 1)$. Rather than choosing the next number $c_{i+1}$ from $(0, c_i)$, we can instead scale $x$ appropriately so that $c_{i+1}$ is chosen from $(0, 1)$, effectively restarting the game with $x/c_i$ instead of $x$.



    To account for the infinitely many ways to choose $c_i in (x, 1)$, we can take the following integral:
    $$textrm{#future moves} = frac1{1-x}int_x^1fleft(frac xuright) du$$



    In other words,
    $$f(x) = 1 + int_x^1fleft(frac xuright) du.tag{1}$$



    The rest (I'm skipping a few steps here...) is a matter of manipulating this to eventually bring it into the form of a differential equation,
    $$xf''(x) = -f'(x),$$
    whose general solution is $f(x) = a + bln x$. From there, using $(1)$ we find $a=1, b=-1$, solving the problem.



    Given the simplicity of the final expression there may be a much more direct solution, but at the moment I don't see one.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Given $x$, let $f(x)$ be the expected length of the game. We'll show that $f(x) = 1 - ln x$.



      The probability that we finish the game on the first step is $x$; the game continues with probability $1 - x$. We therefore have
      $$f(x) = 1 + (1 - x)cdot(textrm{#future moves}).$$



      Now, suppose we have to continue, i.e., $c_i in (x, 1)$. Rather than choosing the next number $c_{i+1}$ from $(0, c_i)$, we can instead scale $x$ appropriately so that $c_{i+1}$ is chosen from $(0, 1)$, effectively restarting the game with $x/c_i$ instead of $x$.



      To account for the infinitely many ways to choose $c_i in (x, 1)$, we can take the following integral:
      $$textrm{#future moves} = frac1{1-x}int_x^1fleft(frac xuright) du$$



      In other words,
      $$f(x) = 1 + int_x^1fleft(frac xuright) du.tag{1}$$



      The rest (I'm skipping a few steps here...) is a matter of manipulating this to eventually bring it into the form of a differential equation,
      $$xf''(x) = -f'(x),$$
      whose general solution is $f(x) = a + bln x$. From there, using $(1)$ we find $a=1, b=-1$, solving the problem.



      Given the simplicity of the final expression there may be a much more direct solution, but at the moment I don't see one.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Given $x$, let $f(x)$ be the expected length of the game. We'll show that $f(x) = 1 - ln x$.



        The probability that we finish the game on the first step is $x$; the game continues with probability $1 - x$. We therefore have
        $$f(x) = 1 + (1 - x)cdot(textrm{#future moves}).$$



        Now, suppose we have to continue, i.e., $c_i in (x, 1)$. Rather than choosing the next number $c_{i+1}$ from $(0, c_i)$, we can instead scale $x$ appropriately so that $c_{i+1}$ is chosen from $(0, 1)$, effectively restarting the game with $x/c_i$ instead of $x$.



        To account for the infinitely many ways to choose $c_i in (x, 1)$, we can take the following integral:
        $$textrm{#future moves} = frac1{1-x}int_x^1fleft(frac xuright) du$$



        In other words,
        $$f(x) = 1 + int_x^1fleft(frac xuright) du.tag{1}$$



        The rest (I'm skipping a few steps here...) is a matter of manipulating this to eventually bring it into the form of a differential equation,
        $$xf''(x) = -f'(x),$$
        whose general solution is $f(x) = a + bln x$. From there, using $(1)$ we find $a=1, b=-1$, solving the problem.



        Given the simplicity of the final expression there may be a much more direct solution, but at the moment I don't see one.






        share|cite|improve this answer









        $endgroup$



        Given $x$, let $f(x)$ be the expected length of the game. We'll show that $f(x) = 1 - ln x$.



        The probability that we finish the game on the first step is $x$; the game continues with probability $1 - x$. We therefore have
        $$f(x) = 1 + (1 - x)cdot(textrm{#future moves}).$$



        Now, suppose we have to continue, i.e., $c_i in (x, 1)$. Rather than choosing the next number $c_{i+1}$ from $(0, c_i)$, we can instead scale $x$ appropriately so that $c_{i+1}$ is chosen from $(0, 1)$, effectively restarting the game with $x/c_i$ instead of $x$.



        To account for the infinitely many ways to choose $c_i in (x, 1)$, we can take the following integral:
        $$textrm{#future moves} = frac1{1-x}int_x^1fleft(frac xuright) du$$



        In other words,
        $$f(x) = 1 + int_x^1fleft(frac xuright) du.tag{1}$$



        The rest (I'm skipping a few steps here...) is a matter of manipulating this to eventually bring it into the form of a differential equation,
        $$xf''(x) = -f'(x),$$
        whose general solution is $f(x) = a + bln x$. From there, using $(1)$ we find $a=1, b=-1$, solving the problem.



        Given the simplicity of the final expression there may be a much more direct solution, but at the moment I don't see one.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 2 '18 at 19:12









        ThéophileThéophile

        19.5k12946




        19.5k12946























            2












            $begingroup$

            This answer uses the following statements:




            Lemma 1: Let $U_1,ldots,U_n$ be independent and identically distributed uniform random variables on the interval $(0,1)$. Then the probability density function of the product $U_1 ldots U_n$ is given by $$ f_n(z) = frac{(-1)^{n-1}}{(n-1)!} log^{n-1}(z) 1_{(0,1)}(z).$$



            Lemma 2: $$int log^{n}(z) , dz = z sum_{k=0}^n (-1)^{n-k} frac{n!}{k!} (log z)^k, qquad n in mathbb{N}$$




            Hints: Let $(U_i)_{i in mathbb{N}}$ be a sequence of independent random variables which are uniformly distributed on $(0,1)$.




            1. Define iteratively $$C_1 := U_1 qquad C_i := U_i C_{i-1}, qquad i geq 2.$$ Check that $(C_i)_{i in mathbb{N}}$ can be used to model the outcome of your probability. Show that $$C_i = prod_{j=1}^i U_j$$ for any $i in mathbb{N}$.

            2. Define $$tau := inf{i in mathbb{N}; C_i < x}.$$ Use the fact that the sequence $C_i$ is strictly increasing in $i$ to show that $$mathbb{P}(tau geq i+1) = mathbb{P}(C_i geq x).$$

            3. By Step 1 and Lemma 1, we have $$mathbb{P}(C_i geq x) = frac{(-1)^{i-1}}{(i-1)!} int_x^1 log^{i-1}(z) , dz$$ Use Lemma 2 to conclude that $$mathbb{P}(C_i geq x) = 1- x sum_{k=0}^{i-1} (-1)^k frac{1}{k!} (log x)^k quad text{for $i geq 2$}.$$


            4. Since $$mathbb{P}(tau=i) = mathbb{P}(tau geq i)-mathbb{P}(tau geq i+1)$$ it follows from Step 2 and 3 that $$mathbb{P}(tau=i) = x (-1)^{i-1} frac{(log x)^{i-1}}{(i-1)!} quad text{for $i geq 3$}.$$ Show that $$mathbb{P}(tau=1) = x qquad mathbb{P}(tau=2) = -x log x$$


            5. Deduce from Step 4 that $$mathbb{E}(tau) = sum_{i in mathbb{N}} i mathbb{P}(tau=i) = x +x sum_{i geq 2} i (-1)^{i-1} frac{log(x)^{i-1}}{(i-1)!}$$


            6. Using $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = frac{d}{dz} sum_{i geq 2} frac{z^i}{(i-1)!}$$ prove that $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = exp(z) (z+1)-1 $$


            7. Conclude that $$mathbb{E}(tau) = 1- log(x).$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
              $endgroup$
              – 0k33
              Dec 3 '18 at 18:44






            • 1




              $begingroup$
              @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
              $endgroup$
              – saz
              Dec 3 '18 at 19:33
















            2












            $begingroup$

            This answer uses the following statements:




            Lemma 1: Let $U_1,ldots,U_n$ be independent and identically distributed uniform random variables on the interval $(0,1)$. Then the probability density function of the product $U_1 ldots U_n$ is given by $$ f_n(z) = frac{(-1)^{n-1}}{(n-1)!} log^{n-1}(z) 1_{(0,1)}(z).$$



            Lemma 2: $$int log^{n}(z) , dz = z sum_{k=0}^n (-1)^{n-k} frac{n!}{k!} (log z)^k, qquad n in mathbb{N}$$




            Hints: Let $(U_i)_{i in mathbb{N}}$ be a sequence of independent random variables which are uniformly distributed on $(0,1)$.




            1. Define iteratively $$C_1 := U_1 qquad C_i := U_i C_{i-1}, qquad i geq 2.$$ Check that $(C_i)_{i in mathbb{N}}$ can be used to model the outcome of your probability. Show that $$C_i = prod_{j=1}^i U_j$$ for any $i in mathbb{N}$.

            2. Define $$tau := inf{i in mathbb{N}; C_i < x}.$$ Use the fact that the sequence $C_i$ is strictly increasing in $i$ to show that $$mathbb{P}(tau geq i+1) = mathbb{P}(C_i geq x).$$

            3. By Step 1 and Lemma 1, we have $$mathbb{P}(C_i geq x) = frac{(-1)^{i-1}}{(i-1)!} int_x^1 log^{i-1}(z) , dz$$ Use Lemma 2 to conclude that $$mathbb{P}(C_i geq x) = 1- x sum_{k=0}^{i-1} (-1)^k frac{1}{k!} (log x)^k quad text{for $i geq 2$}.$$


            4. Since $$mathbb{P}(tau=i) = mathbb{P}(tau geq i)-mathbb{P}(tau geq i+1)$$ it follows from Step 2 and 3 that $$mathbb{P}(tau=i) = x (-1)^{i-1} frac{(log x)^{i-1}}{(i-1)!} quad text{for $i geq 3$}.$$ Show that $$mathbb{P}(tau=1) = x qquad mathbb{P}(tau=2) = -x log x$$


            5. Deduce from Step 4 that $$mathbb{E}(tau) = sum_{i in mathbb{N}} i mathbb{P}(tau=i) = x +x sum_{i geq 2} i (-1)^{i-1} frac{log(x)^{i-1}}{(i-1)!}$$


            6. Using $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = frac{d}{dz} sum_{i geq 2} frac{z^i}{(i-1)!}$$ prove that $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = exp(z) (z+1)-1 $$


            7. Conclude that $$mathbb{E}(tau) = 1- log(x).$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
              $endgroup$
              – 0k33
              Dec 3 '18 at 18:44






            • 1




              $begingroup$
              @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
              $endgroup$
              – saz
              Dec 3 '18 at 19:33














            2












            2








            2





            $begingroup$

            This answer uses the following statements:




            Lemma 1: Let $U_1,ldots,U_n$ be independent and identically distributed uniform random variables on the interval $(0,1)$. Then the probability density function of the product $U_1 ldots U_n$ is given by $$ f_n(z) = frac{(-1)^{n-1}}{(n-1)!} log^{n-1}(z) 1_{(0,1)}(z).$$



            Lemma 2: $$int log^{n}(z) , dz = z sum_{k=0}^n (-1)^{n-k} frac{n!}{k!} (log z)^k, qquad n in mathbb{N}$$




            Hints: Let $(U_i)_{i in mathbb{N}}$ be a sequence of independent random variables which are uniformly distributed on $(0,1)$.




            1. Define iteratively $$C_1 := U_1 qquad C_i := U_i C_{i-1}, qquad i geq 2.$$ Check that $(C_i)_{i in mathbb{N}}$ can be used to model the outcome of your probability. Show that $$C_i = prod_{j=1}^i U_j$$ for any $i in mathbb{N}$.

            2. Define $$tau := inf{i in mathbb{N}; C_i < x}.$$ Use the fact that the sequence $C_i$ is strictly increasing in $i$ to show that $$mathbb{P}(tau geq i+1) = mathbb{P}(C_i geq x).$$

            3. By Step 1 and Lemma 1, we have $$mathbb{P}(C_i geq x) = frac{(-1)^{i-1}}{(i-1)!} int_x^1 log^{i-1}(z) , dz$$ Use Lemma 2 to conclude that $$mathbb{P}(C_i geq x) = 1- x sum_{k=0}^{i-1} (-1)^k frac{1}{k!} (log x)^k quad text{for $i geq 2$}.$$


            4. Since $$mathbb{P}(tau=i) = mathbb{P}(tau geq i)-mathbb{P}(tau geq i+1)$$ it follows from Step 2 and 3 that $$mathbb{P}(tau=i) = x (-1)^{i-1} frac{(log x)^{i-1}}{(i-1)!} quad text{for $i geq 3$}.$$ Show that $$mathbb{P}(tau=1) = x qquad mathbb{P}(tau=2) = -x log x$$


            5. Deduce from Step 4 that $$mathbb{E}(tau) = sum_{i in mathbb{N}} i mathbb{P}(tau=i) = x +x sum_{i geq 2} i (-1)^{i-1} frac{log(x)^{i-1}}{(i-1)!}$$


            6. Using $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = frac{d}{dz} sum_{i geq 2} frac{z^i}{(i-1)!}$$ prove that $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = exp(z) (z+1)-1 $$


            7. Conclude that $$mathbb{E}(tau) = 1- log(x).$$






            share|cite|improve this answer











            $endgroup$



            This answer uses the following statements:




            Lemma 1: Let $U_1,ldots,U_n$ be independent and identically distributed uniform random variables on the interval $(0,1)$. Then the probability density function of the product $U_1 ldots U_n$ is given by $$ f_n(z) = frac{(-1)^{n-1}}{(n-1)!} log^{n-1}(z) 1_{(0,1)}(z).$$



            Lemma 2: $$int log^{n}(z) , dz = z sum_{k=0}^n (-1)^{n-k} frac{n!}{k!} (log z)^k, qquad n in mathbb{N}$$




            Hints: Let $(U_i)_{i in mathbb{N}}$ be a sequence of independent random variables which are uniformly distributed on $(0,1)$.




            1. Define iteratively $$C_1 := U_1 qquad C_i := U_i C_{i-1}, qquad i geq 2.$$ Check that $(C_i)_{i in mathbb{N}}$ can be used to model the outcome of your probability. Show that $$C_i = prod_{j=1}^i U_j$$ for any $i in mathbb{N}$.

            2. Define $$tau := inf{i in mathbb{N}; C_i < x}.$$ Use the fact that the sequence $C_i$ is strictly increasing in $i$ to show that $$mathbb{P}(tau geq i+1) = mathbb{P}(C_i geq x).$$

            3. By Step 1 and Lemma 1, we have $$mathbb{P}(C_i geq x) = frac{(-1)^{i-1}}{(i-1)!} int_x^1 log^{i-1}(z) , dz$$ Use Lemma 2 to conclude that $$mathbb{P}(C_i geq x) = 1- x sum_{k=0}^{i-1} (-1)^k frac{1}{k!} (log x)^k quad text{for $i geq 2$}.$$


            4. Since $$mathbb{P}(tau=i) = mathbb{P}(tau geq i)-mathbb{P}(tau geq i+1)$$ it follows from Step 2 and 3 that $$mathbb{P}(tau=i) = x (-1)^{i-1} frac{(log x)^{i-1}}{(i-1)!} quad text{for $i geq 3$}.$$ Show that $$mathbb{P}(tau=1) = x qquad mathbb{P}(tau=2) = -x log x$$


            5. Deduce from Step 4 that $$mathbb{E}(tau) = sum_{i in mathbb{N}} i mathbb{P}(tau=i) = x +x sum_{i geq 2} i (-1)^{i-1} frac{log(x)^{i-1}}{(i-1)!}$$


            6. Using $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = frac{d}{dz} sum_{i geq 2} frac{z^i}{(i-1)!}$$ prove that $$ sum_{i geq 2} i frac{z^{i-1}}{(i-1)!} = exp(z) (z+1)-1 $$


            7. Conclude that $$mathbb{E}(tau) = 1- log(x).$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 3 '18 at 7:49

























            answered Dec 2 '18 at 19:59









            sazsaz

            79k858123




            79k858123












            • $begingroup$
              thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
              $endgroup$
              – 0k33
              Dec 3 '18 at 18:44






            • 1




              $begingroup$
              @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
              $endgroup$
              – saz
              Dec 3 '18 at 19:33


















            • $begingroup$
              thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
              $endgroup$
              – 0k33
              Dec 3 '18 at 18:44






            • 1




              $begingroup$
              @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
              $endgroup$
              – saz
              Dec 3 '18 at 19:33
















            $begingroup$
            thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
            $endgroup$
            – 0k33
            Dec 3 '18 at 18:44




            $begingroup$
            thank you so much for your answer and detailed explanation :) I just wanted to let you know that I am interested in collecting other types of solutions, too, which is why I haven't marked any answer as accepted yet.
            $endgroup$
            – 0k33
            Dec 3 '18 at 18:44




            1




            1




            $begingroup$
            @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
            $endgroup$
            – saz
            Dec 3 '18 at 19:33




            $begingroup$
            @0k33 Yeah, sure, no problem. (I doubt that there is much more to collect but perhaps I'm wrong. )
            $endgroup$
            – saz
            Dec 3 '18 at 19:33


















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