What is the moment-generating function of a normal random variable?

What is the moment-generating function of a normal random variable?

(8) The moment generating function corresponding to the normal probability density function N(x;µ, σ2) is the function Mx(t) = exp{µt + σ2t2/2}.

Does the moment-generating function characterize a distribution?

The distribution of a random variable is often characterized in terms of its moment generating function (mgf), a real function whose derivatives at zero are equal to the moments of the random variable.

What is uniqueness property of MGF?

uniquely defines the distribution of a random variable. That is, if you can show that the moment generating function of is the same as some known moment-generating function, then follows the same distribution.

Which is the moment generating function of the normal distribution?

The Moment Generating Function of the Normal Distribution Suppose X is normal with mean 0 and standard deviation 1. Then its moment generating function is: M(t) =E

Which is the moment generating function of the mean?

The moment generating function of the sample mean X ¯ = ∑ i = 1 n ( 1 n) X i is M X ¯ ( t) = ∏ i = 1 n M ( t n) = [ M ( t n)] n. Let X 1, X 2, and X 3 denote a random sample of size 3 from a gamma distribution with α = 7 and θ = 5. Let Y be the sum of the three random variables: What is the distribution of Y?

Is the moment generating function Y A binomial variable?

That is, Y has the same moment-generating function as a binomial random variable with n = 5 and p = 1 2. Therefore, by the uniqueness properties of moment-generating functions, Y must be a binomial random variable with n = 5 and p = 1 2. (Of course, we already knew that!)

How to find the distribution of a random variable?

Recall that the moment generating function: uniquely defines the distribution of a random variable. That is, if you can show that the moment generating function of X ¯ is the same as some known moment-generating function, then X ¯ follows the same distribution. So, one strategy to finding the distribution of a function of random variables is: