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cutesteph
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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 .
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.
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.
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.
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.
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.