Question

What is the significance of the mean of a probability​ distribution?

Answer 1

Answer:

See explanation below.

Step-by-step explanation:

If we have a continuous variable the expected value is defined as:

$E(X) = \mu = \int_{a}^b x f(x) dx$

Where a and b are the limits for the distribution and $f(x)$ represent the density function.

If we have a discrete random variable X, the expected value is defined as:

$E(X) = \sum_{i=1}^n x_i P(X_i)$

The mean is the most common measure of central tendency in order to describe a probability distribution.

The expected value also represent the first central moment of the random variable defined as:

$\mu_1= E[(X-E[X])^1] =\int_{-\infty}^{\infty} (x-\mu)^n f(x) dx$

If we assum that X is a continuous random variable.