Finding an upper bound for a probability

[sabbagh80]

Hi,

I will thank If somebody help me solving this problem.

Consider a random variable k_1 with the given pmf as:

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

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

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

For example; we can use the inequality of 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}, but it is not that much tight.

Can everybody help me?

 

[mathman]

Your latex isn't working!

 

[sabbagh80]

Hi,

I will thank If somebody help me solving this problem.

Consider a random variable k_1 with the given pmf as:

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}


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

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

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

For example; we can use the inequality of 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}, but it is not that much tight.

Can everybody help me?