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Finding an upper bound for a probability

  1. Jul 13, 2011 #1
    Hi,

    I will thank If somebody help me solving this problem.

    Consider a random variable [itex] k_1 [/itex] with the given pmf as:

    [itex]Pr[k_1=l]=\sum_{l_1+2l_2=l} \frac{N!}{(N-l_1-l_2)!l_1!l_2!}p_1^{l_1} p_2^{l_2} (1-(p_1+p_2))^{N-l_1-l_2}[/itex]


    where [itex]l_1,l_2 \in [0,1,...,l] [/itex].

    but we don't have [itex]p_1[/itex] and [itex]p_2[/itex] separately and I know just the value of [itex]p_1+p_2[/itex].

    I want to find at least a good and tight upper bound for the above pmf.

    For example; we can use the inequality of [itex] p_1^{l_1} p_2^{l_2} \leq \frac{l_1!l_2!}{(l_1+l_2)!}(p_1+p_2)^{l_1+l_2} [/itex], but it is not that much tight.

    Can everybody help me?
     
    Last edited: Jul 13, 2011
  2. jcsd
  3. Jul 13, 2011 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Your latex isn't working!
     
  4. Jul 14, 2011 #3
    Hi,

    I will thank If somebody help me solving this problem.

    Consider a random variable [itex] k_1 [/itex] with the given pmf as:

    [itex]Pr[k_1=l]=\sum_{l_1+2l_2=l} \frac{N!}{(N-l_1-l_2)!l_1!l_2!}p_1^{l_1} p_2^{l_2} (1-(p_1+p_2))^{N-l_1-l_2}[/itex]


    where [itex]l_1,l_2 \in [0,1,...,l] [/itex].

    but we don't have [itex]p_1[/itex] and [itex]p_2[/itex] separately and I know just the value of [itex]p_1+p_2[/itex].

    I want to find at least a good and tight upper bound for the above pmf.

    For example; we can use the inequality of [itex] p_1^{l_1} p_2^{l_2} \leq \frac{l_1!l_2!}{(l_1+l_2)!}(p_1+p_2)^{l_1+l_2} [/itex], but it is not that much tight.

    Can everybody help me?
     
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