# Finding an upper bound for a probability

• sabbagh80
In summary, the problem at hand is to find a good and tight upper bound for a given pmf of a random variable k_1, where the values of p_1 and p_2 are unknown and only the sum of the two is known. Various approaches, such as using inequalities, have been proposed but none have provided a satisfactory solution. Assistance in solving this problem is requested.
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:

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?

## 1. What is an upper bound for a probability?

An upper bound for a probability is the maximum possible value that a probability can reach. It is a theoretical limit that the probability cannot exceed.

## 2. Why is it important to find an upper bound for a probability?

Finding an upper bound for a probability is important because it helps us understand the maximum likelihood of an event occurring, and can help us make decisions or predictions based on that information. It also helps us analyze and compare different probability distributions.

## 3. How can an upper bound for a probability be determined?

An upper bound for a probability can be determined by analyzing the characteristics of the event or experiment, such as the sample space, the number of possible outcomes, and the likelihood of each outcome. It can also be calculated using mathematical methods, such as the Chebyshev's inequality or Markov's inequality.

## 4. Can an upper bound for a probability ever be equal to the actual probability?

No, an upper bound for a probability is always greater than or equal to the actual probability. This means that the actual probability can be equal to the upper bound, but it cannot exceed it.

## 5. Are there any limitations to finding an upper bound for a probability?

Yes, there are limitations to finding an upper bound for a probability. For example, it may not accurately represent the true probability in real-world scenarios where there are many unknown variables. It also cannot account for extreme or rare events that may have a low probability of occurring but still have a significant impact.

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