What is the moment generating function of the random variable X?

What is the moment generating function of the random variable X?

The moment generating function (MGF) of a random variable X is a function MX(s) defined as MX(s)=E[esX]. We say that MGF of X exists, if there exists a positive constant a such that MX(s) is finite for all s∈[−a,a].

What is the mean of a uniform distribution with parameters A and B?

The Uniform distribution is a univariate continuous distribution. The standard uniform distribution has parameters a = 0 and b = 1 resulting in f(t) = 1 within a and b and zero elsewhere.

What is the expected value of x E x )?

The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.

What is the moment-generating function of uniform distribution?

The moment-generating function is: For a random variable following this distribution, the expected value is then m1 = (a + b)/2 and the variance is m2 − m12 = (b − a)2/12.

What is moment-generating function in statistics?

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.

What is the moment generating function of uniform distribution?

How do you find the distribution function of a uniform distribution?

The general formula for the probability density function (pdf) for the uniform distribution is: f(x) = 1/ (B-A) for A≤ x ≤B.

What does var x mean in statistics?

A measure of spread for a distribution of a random variable that determines the degree to which the values of a random variable differ from the expected value. The variance of random variable X is often written as Var(X) or σ2 or σ2x.

How do you find the moment generating function of a continuous random variable?

It is easy to show that the moment generating function of X is given by etμ+(σ2/2)t2 . Now suppose that X and Y are two independent normal random variables with parameters μ1, σ1, and μ2, σ2, respectively. Then, the product of the moment generating functions of X and Y is et(μ1+μ2)+((σ21+σ22)/2)t2 .

How to find the moment generating function of a random variable?

Moment generating functions (mgfs) are function of t. You can find the mgfs by using the definition of expectation of function of a random variable. The moment generating function of X is M X (t) = E [ e t X] = E [ exp (t X)]

What is the moment generating function of T?

Moment generating functions (mgfs) are function of t. You can find the mgfs by using the definition of expectation of function of a random variable. The moment generating function of X is Besides helping to find moments, the moment generating function has an important property often called the uniqueness property.

What is a moment in statistics?

The expected values E ( X), E ( X 2), E ( X 3), …, and E ( X r) are called moments. As you have already experienced in some cases, the mean: which are functions of moments, are sometimes difficult to find. Special functions, called moment-generating functions can sometimes make finding the mean and variance of a random variable simpler.

What is the uniqueness property of moment generating function?

Besides helping to find moments, the moment generating function has an important property often called the uniqueness property. The uniqueness property means that, if the mgf exists for a random variable, then there one and only one distribution associated with that mgf.