What is moment generating function of a random variable?

What is moment generating function of a random variable?

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]. Before going any further, let’s look at an example.

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 .

What is the moment generating function of discrete uniform distribution?

Let X be a discrete random variable with a discrete uniform distribution with parameter n for some n∈N. Then the moment generating function MX of X is given by: MX(t)=et(1−ent)n(1−et)

What is PDF in statistics?

What Is a Probability Density Function (PDF)? Probability density function (PDF) is a statistical expression that defines a probability distribution (the likelihood of an outcome) for a discrete random variable (e.g., a stock or ETF) as opposed to a continuous random variable.

What is the PDF 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. “A” is the location parameter: The location parameter tells you where the center of the graph is.

What is uniform distribution example?

A deck of cards also has a uniform distribution. It is because an individual has an equal chance of drawing a spade, a heart, a club, or a diamond. Another example of a uniform distribution is when a coin is tossed. The likelihood of getting a tail or head is the same.

What is a moment of function?

In mathematics, the moments of a function are quantitative measures related to the shape of the function’s graph. If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia.

What is PMF PDF and CDF?

PMF uses discrete random variables. PDF uses continuous random variables. Based on studies, PDF is the derivative of CDF, which is the cumulative distribution function. CDF is used to determine the probability wherein a continuous random variable would occur within any measurable subset of a certain range.

What is a uniform random variable?

1 The Uniform Random Variable. A random variable is said to be uniformly distributed over the interval if its probability density function is given by. f ( x ) = 1 , 0 < x < 1 0 , otherwise. Note that the preceding is a density function since f ( x ) ≥ 0 and. ∫ – ∞ ∞ f ( x ) dx = ∫ 0 1 dx = 1.