What do these new symbols mean ?I can't start this without knowing

  1. 1. The problem statement, all variables and given/known data


    [​IMG]



    3. The attempt at a solution

    What is the [tex]P^n _{k}[/tex] part thing?

    Someone should probably go over the i) for me too...
     
  2. jcsd
  3. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    The [itex]P_n^k[/itex] is just a symbol. It is a name for a polynomial defined by

    [tex]P_n^k(x)=\binom{n}{k}x^k(1-x)^{n-k}[/tex]

    So it's just like we define [itex]P(x)=x^2[/itex]. But now our polynomial depends of n and k.
     
  4. For the i)

    [tex]\sum^{n}_{k=0} \left( \alpha f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} + \beta g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}\right) = \alpha \sum^{n}_{k=0} f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} + \beta \sum^{n}_{k=0} g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}[/tex]

    Summation property??
     
  5. Something is wrong...do I need induction? This was from proof class
     
  6. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    For (i), you need to start from

    [tex]B_n(\alpha f+\beta g)=\sum_{k=0}^n{(\alpha f+\beta g)(\frac{k}{n})x^k(1-x)^{n-k}}[/tex]
     
  7. What's wrong with what I did...? GO on grill me!!
     
  8. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    Nothing is wrong, it's all correct, but it's not finished yet.

    You still need to show that the left-hand-side of your post equals

    [tex]B_n(\alpha f+\beta g)[/tex]

    and that the right-hand-side equals

    [tex]\alpha B_n(f)+\beta B_n(g)[/tex]
     
  9. [tex]B_n(\alpha f+\beta g)=\sum_{k=0}^n{(\alpha f+\beta g)(\frac{k}{n})x^k(1-x)^{n-k}} [/tex]

    [tex]= \sum_{k=0}^n{(\alpha f (\frac{k}{n})x^k(1-x)^{n-k}} + \beta g(\frac{k}{n})x^k(1-x)^{n-k} ) = \sum_{k=0}^n{\alpha f (\frac{k}{n})x^k(1-x)^{n-k}} + \sum_{k=0}^n \beta g(\frac{k}{n})x^k(1-x)^{n-k} ) = \alpha B_n(f)+\beta B_n(g)[/tex]

    okay done, got lazy with the long tex
     
  10. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    Yeah, that looks good!!
     
  11. How do I start ii) then?
     
  12. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    You need to prove

    [tex]B_n(f)\leq B_n(g)[/tex]

    Start by writing these things out according to the definition of the [itex]B_n[/itex].
     
  13. [tex] \sum^{n}_{k=0} f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} \leq \sum^{n}_{k=0} g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}/tex]

    SOme kinda of cancellations...?
     
  14. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    Well, you know that

    [tex]f(k/n)\leq g(k/n)[/tex]

    Now try to introduce the terms needed to conclude that [itex]B_nf\leq B_ng[/itex].
     
  15. Can you multiply P to both sides...? I don't know if P is always positive
     
  16. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    That's the idea.

    Try to prove it then. Prove that [itex]P_k^n(x)[/itex] is positive if [itex]x\in [0,1][/itex]...
     
  17. So I have to prove two things...

    OKay since [tex]x \in [0, 1] \leq 0[/tex], then it doesn't matter what i put in right? Now how do do that in proper English...?
     
  18. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    This makes no sense to me...
     
  19. I meant to say [tex]x \in [0, 1] \geq 0[/tex]

    sorry
     
  20. micromass

    micromass 19,347
    Staff Emeritus
    Science Advisor
    Education Advisor

    Yeah, that also makes no sense. How can [itex][0,1]\geq 0[/itex]?? [0,1] is a set.
     
  21. OKay I wanted to say that numbers in [0,1] are alll positive, so we never had to worry about odd powers messing up with negative numbers
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?