Finding an upper bound for a probability

  • Thread starter sabbagh80
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  • #1
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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:

Answers and Replies

  • #2
mathman
Science Advisor
7,858
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Your latex isn't working!
 
  • #3
38
0
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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