# Finding an upper bound for a probability

1. Jul 13, 2011

### 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?

Last edited: Jul 13, 2011
2. Jul 13, 2011

### mathman

3. Jul 14, 2011

### 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?