# Proof of mean and variance of standard normal distribution pdf

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- Normal distribution
- Lesson 16: Normal Distributions
- Standard Normal Distribution
- Normal distribution

## Normal distribution

We have seen that for a discrete random variable, that the expected value is the sum of all xP x. For continuous random variables, P x is the probability density function, and integration takes the place of addition. Let f x be a probability density function on the domain [a,b] , then the expected value of f x is. We use integration by parts with. We have. The variance formula for a continuous random variable also follows from the variance formula for a discrete random variable. Once again we interpret the sum as an integral.

## Lesson 16: Normal Distributions

Exploratory Data Analysis 1. EDA Techniques 1. Probability Distributions 1. Gallery of Distributions 1. The following is the plot of the standard normal probability density function. It is computed numerically. The following is the plot of the normal cumulative distribution function.

The multivariate normal distribution is among the most important of multivariate distributions, particularly in statistical inference and the study of Gaussian processes such as Brownian motion. The distribution arises naturally from linear transformations of independent normal variables. In this section, we consider the bivariate normal distribution first, because explicit results can be given and because graphical interpretations are possible. Then, with the aid of matrix notation, we discuss the general multivariate distribution. The basic properties of the standard bivariate normal distribution follow easily from independence and properties of the univariate normal distribution. Parts a and b are clear.

These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes. Random variables can be discrete or continuous. A basic function to draw random samples from a specified set of elements is the function sample , see? We can use it to simulate the random outcome of a dice roll. The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value.

## Standard Normal Distribution

Normal distribution , also called Gaussian distribution , the most common distribution function for independent, randomly generated variables. Its familiar bell-shaped curve is ubiquitous in statistical reports, from survey analysis and quality control to resource allocation. The graph of the normal distribution is characterized by two parameters: the mean , or average, which is the maximum of the graph and about which the graph is always symmetric; and the standard deviation , which determines the amount of dispersion away from the mean. A small standard deviation compared with the mean produces a steep graph, whereas a large standard deviation again compared with the mean produces a flat graph.

### Normal distribution

The normal distribution refers to a family of continuous probability distributions described by the normal equation. The random variable X in the normal equation is called the normal random variable. The normal equation is the probability density function for the normal distribution. The graph of the normal distribution depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. All normal distributions look like a symmetric, bell-shaped curve, as shown below.

Data are said to be normally distributed if their frequency histogram is apporximated by a bell shaped curve. In practice, one can tell by looking at a histogram if the data are normally distributed. The bell shaped curve was discovered by Carl Friedrich Gauss , whom many mathematical historians consider to have been the greatest mathematician of all time. Gauss was working as the royal surveyor for the king of Prussia. Surveyors maesure distances. For instance, a survey crew may measure a distance to be To tell if that is the correct distance, they would check their work by measuring it again.

So far we have looked at expected value, standard deviation, and variance for discrete random Since the probability density increases as x increases over the range, the The standard normal distribution is symmetric and has mean 0.