1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probabilities inequalities

  1. Jun 7, 2013 #1
    the random value X has the inequality , -10<=X<=10 and E(X)=2, what is the minimum upper bound
    of the probability P(X>=5) ?


    my first thought was to find this P(X>=t)<=E(X)/t which is 2/5 from Markov but its not correct, any ideas?
     
  2. jcsd
  3. Jun 7, 2013 #2

    verty

    User Avatar
    Homework Helper

    My very crude idea is to set all the weights to zero and then choose two weights only to adjust. I won't say more.
     
  4. Jun 7, 2013 #3
    weights? what do you mean?
     
  5. Jun 7, 2013 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Why do you say 2/5 is wrong? It is not wrong, because for any number p (0 ≤ p ≤ 2/5) we can find a random variable X that satisfies the conditions and gives P(X ≥ 5) = p. So, no number less than 2/5 can possibly be an UPPER bound, because if we take a number p with p < 2/5 we can always find a suitable X having P(X ≥ 5) > p (but ≤ 2/5); in fact, we can find infinitely many suitable X having P(X ≥ 5) = 2/5 exactly. On the other hand, 2/5 is certainly an upper bound, because no suitable X can have a probability P(X ≥ 5) that exceeds 2/5---that's what Markov's inequality is all about.

    I will leave it to you to verify the statements I have made; you do need to verify them, since otherwise your "solution" would be that 'somebody said so', and that is not a proof.
     
  6. Jun 7, 2013 #5
    yea but why X is -10<X<10 ? this made me to think that i must find other bounds upper...
     
  7. Jun 7, 2013 #6

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Consider distributions which can take two values only.
    This is the same as "the weight is non-zero for two values only", verty's idea.
     
  8. Jun 7, 2013 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Which post are your responding to? Use the proper 'reply' buttons; otherwise, it is impossible to tell what part of the thread your post addresses.
     
  9. Jun 7, 2013 #8

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Isn't that predicated on X>=0?
     
  10. Jun 7, 2013 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Yes, sorry. Disregard my silly posting; I've been dizzy and feverish for most of today and I should have stayed in bed.
     
  11. Jun 7, 2013 #10

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Sorry to hear that - hope you feel better soon.

    ParisSpart, in order to make use of the Markov inequality you will need to map X to a random variable which is always >= 0.
     
  12. Jun 7, 2013 #11

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Better now.
     
  13. Jun 8, 2013 #12
    how i will find X>=0 , i can find that abs(X)=<10 what i can take from this?
     
  14. Jun 8, 2013 #13

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    No, X can be < 0, so I'm suggesting creating a new random variable Y as a function of X (the simpler the better) which satisfies Y >= 0. P[X>=5] will equal P[Y>=c] for some c. You can apply LMVT to Y.
     
  15. Jun 8, 2013 #14
    What do you mean with LMVT?
     
  16. Jun 8, 2013 #15

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Sorry, mixing up threads. I meant Markov's inequality.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted