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The formula is given as E(X)=?=?xP(x). Here x represents values of the random variable X, P(x), represents the corresponding probability, and symbol ? represents the sum of all products xP(x). Here we use symbol ? for the mean because it is a parameter. It represents the mean of a population.

If X has this normal distribution, we write. X ? N(µ, ?2. ). This notation says ?X is normally distributed with mean µ and variance. ?2 .? We call µ and ?2 the parameters of the normal distribution.

Sampling distribution of the mean, sampling distribution of proportion, and T-distribution are three major types of finite-sample distribution. The central limit theorem states how the distribution still remains normal and almost accurate with increasing sample size.

In particular, the empirical rule predicts that in normal distributions, 68% of observations fall within the first standard deviation (µ ± ?), 95% within the first two standard deviations (µ ± 2?), and 99.7% within the first three standard deviations (µ ± 3?) of the mean.

A continuous random variable Z is said to be a standard normal (standard Gaussian) random variable, shown as Z?N(0,1), if its PDF is given by fZ(z)=1?2?exp{?z22},for all z?R. The 1?2? is there to make sure that the area under the PDF is equal to one. We will verify that this holds in the solved problems section.

Properties It is symmetric. A normal distribution comes with a perfectly symmetrical shape. ... The mean, median, and mode are equal. The middle point of a normal distribution is the point with the maximum frequency, which means that it possesses the most observations of the variable. ... Empirical rule. ... Skewness and kurtosis.

Standard deviation in statistics, typically denoted by ?, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean (or expected value), ?.

The standard normal distribution is a normal distribution with mean ? = 0 and standard deviation ? = 1. The letter Z is often used to denote a random variable that follows this standard normal distribution.