# Does This Probabilistic Inequality Hold for IID Random Variables?

• forumfann
In summary, the conversation discusses whether a certain inequality holds for non-negative and identically-distributed random variables, with the question of whether it still holds without the independence assumption. It is suggested to consider cases separately and it is mentioned that the inequality does hold if the number 2 is changed to 3. The question is then raised about whether the inequality holds for Bernoulli variables.
forumfann
Suppose x_1,x_2,x_3,x_4 are non-negative Independent and identically-distributed random variables, is it true that $$P\left(x_{1}+x_{2}+x_{3}+x_{4}<2\delta\right)\leq2P\left(x_{1}<\delta\right)$$ for any $$\delta>0$$?

Any answer or suggestion will be highly appreciated!

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This might well hold without the independence assumption. Use x1+x2+x3+x4>=x1+x2 then consider the cases x1<d and x1>=d separately.

Thanks. But then is it true that $$P\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}<3\delta\right)\leq2P\left(x_{1}<\delta\right)$$ for any $$\delta>0$$ ?

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forumfann said:
Thanks. But then is it true that $$P\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}<3\delta\right)\leq2P\left(x_{1}<\delta\right)$$ for any $$\delta>0$$ ?

This is not easy. Change the 2 to 3 and it is certainly true (using same method as before). What if the variables are Bernoulli, does the inequality hold?

## 1. What is a probabilistic inequality?

A probabilistic inequality is a mathematical statement that describes the relationship between two or more random variables. It is used to quantify the likelihood of one variable being greater or less than another variable.

## 2. How is a probabilistic inequality different from a regular inequality?

Unlike a regular inequality, which compares two fixed values, a probabilistic inequality compares random variables that can take on different values with varying probabilities.

## 3. What are some examples of probabilistic inequalities?

Some examples of probabilistic inequalities include the Chebyshev's inequality, the Markov's inequality, and the Chernoff bound. These inequalities are commonly used in statistics and probability theory to make predictions about the distribution of random variables.

## 4. How can probabilistic inequalities be used in scientific research?

Probabilistic inequalities are useful in scientific research for making predictions and drawing conclusions based on data that is subject to uncertainty. They can also be used to set bounds on the likelihood of certain events occurring, which can be helpful in risk assessment and decision making.

## 5. Are there any limitations to using probabilistic inequalities?

Like any mathematical tool, there are limitations to using probabilistic inequalities. They are based on certain assumptions and may not always accurately reflect real-world situations. It is important to carefully consider the context and underlying assumptions before applying a probabilistic inequality in research.

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