Inequality of MGF: Show p(tX >s^2 +logM(t)) < e^-s^2

Thread starter cutesteph
Start date Nov 3, 2012
Tags

Inequality

In summary, "inequality of MGF" refers to a comparison between the moment generating functions of two random variables. The expression "p(tX >s^2 +logM(t))" represents the probability of a random variable being greater than a certain threshold. The inequality "p(tX >s^2 +logM(t)) < e^-s^2" is derived from the Markov inequality and tells us that the MGF can be used as a bound for the probability of a random variable being greater than a certain value. This inequality can be applied to any type of random variable, as long as the MGF exists for that variable. However, it may not hold for all values of t and s.

Nov 3, 2012

cutesteph

MGF of X denote M(x)=E[exp(tx)] exists for every t>0 . For t>0 Show p(tX >s^2 +logM(t)) < e^-s^2 .

Physics news on Phys.org

Nov 3, 2012

chiro

Science Advisor

Hey cutesteph.

Does Chebychev's inequality work here?

Nov 4, 2012

cutesteph

Using chebychev's inequality P( | x-u | >= K(sigma) )=< 1/k^2

x=exp(tx) u= M(t) k=exp(s) sigma=exp(s) Is this correct? Why is the variance exp(s)?

1. What is the meaning of "inequality of MGF" in this context?

Inequality of MGF refers to the inequality involving the moment generating function (MGF) of a random variable. The MGF is a mathematical function that uniquely characterizes the probability distribution of a random variable, and this inequality is a way to compare two MGFs.

2. What does the expression "p(tX >s^2 +logM(t))" represent?

This expression represents the probability that a random variable, tX, is greater than a certain value, s^2 + logM(t). In other words, it is the probability of an event where the random variable has a value greater than a specified threshold.

3. How is the inequality "p(tX >s^2 +logM(t)) < e^-s^2" derived?

This inequality is derived from the Markov inequality, which states that for a non-negative random variable X and any positive number a, the probability of X being greater than or equal to a is less than or equal to the expected value of X divided by a. By setting X to be tX and a to be e^s^2, we can derive the inequality in question.

4. What does this inequality tell us about the relationship between the MGF and the probability of a random variable being greater than a certain value?

This inequality tells us that the MGF can be used as a bound for the probability of a random variable being greater than a certain value. In other words, it provides an upper limit for the probability of an event occurring.

5. Can this inequality be applied to any type of random variable?

Yes, this inequality is applicable to any type of random variable as long as the MGF exists for that variable. However, the inequality may not hold for all values of t and s, as it depends on the properties of the specific MGF being compared.