Normal distribution

Standard normal distribution

The standard normal distribution is the Normal distribution with mean and standard deviation .

It has the simplified probability density function

Due to its symmetry about ,

Standardisation

Any normally distributed variable can be standardised using the Location-scale transformation

where the standardised version of a value is sometimes called the Z-score. This allows the calculation of probabilities for any normally-distributed random variable using the Cumulative distribution function for the standard normal distribution .

Properties

Let .

1. Moment-generating function:
2. Moments:
3. Excess kurtosis: 0

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