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If the moment generating function for the random variable X is M[X(t)] = 1/(1+t), what is the third moment of X about the point x = 2? The general formula only states how to find moments about x = 0. Thanks!
andrewkirk said:The n-th moment of X about a is defined as ##E[(X-a)^n]##.
Substitute n=3 and a=2 into that, then expand ##(X-a)^n## to polynomial form and substitute that into the above expression. You will get the sum of a bunch of moments about zero, which you can then use the MGF to calculate.
A moment generating function is a function that is used to uniquely describe the probability distribution of a random variable. It is a mathematical tool that allows us to calculate moments, such as the mean and variance, of a distribution.
The moment generating function is related to moments of a distribution through its derivatives. The k-th derivative of the moment generating function evaluated at 0 gives the k-th moment of the distribution. This means that the moment generating function contains all the information about the moments of a distribution.
A moment generating function allows us to simplify calculations involving moments of a distribution. It also allows us to easily find the distribution of a sum of independent random variables, by simply multiplying their individual moment generating functions.
In hypothesis testing, the moment generating function is used to calculate the test statistic, which is then compared to a critical value to determine the significance of the results. This allows us to make statistical inferences about a population based on a sample.
No, a moment generating function can only be used for distributions that have finite moments. In other words, the moment generating function exists only if the moments of the distribution exist, which is not the case for all distributions. However, for most commonly used distributions, such as the normal, binomial, and exponential distributions, the moment generating function exists.