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Josh S Thompson
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How do you know that if two random variables have the same moment generating function then they have the same probability distribution.
https://en.wikipedia.org/wiki/Moment-generating_functionJosh S Thompson said:How do you know that if two random variables have the same moment generating function then they have the same probability distribution.
A moment generating function (MGF) is a mathematical function that is used to describe the probability distribution of a random variable. It is defined as the expected value of e^tx, where t is the variable and x is the random variable.
The moment generating function uniquely determines the probability distribution of a random variable. This means that if two random variables have the same moment generating function, they must also have the same probability distribution.
If two random variables have the same moment generating function, it means that they have the same probability distribution. This is important because it allows us to use the properties and formulas of the moment generating function to calculate probabilities and other statistical measures for both variables.
The moments of a distribution can be calculated by taking the derivatives of the moment generating function at t=0. The first derivative gives the mean, the second derivative gives the variance, and so on. This makes the moment generating function a useful tool for finding moments without having to use complicated integration techniques.
Yes, the moment generating function can be used for all types of probability distributions, as long as the expected value of e^tx exists. This includes discrete and continuous distributions, as well as both symmetric and asymmetric distributions.