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The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve, as it allows for the standardization of different normal distributions.

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In the context of the standard normal distribution, the z-score helps determine the area under the curve to the left of a specific value, which corresponds to the probability of a random variable being less than that value.

The area under the curve (AUC) in a probability distribution represents the likelihood of a random variable falling within a certain range. For the standard normal distribution, this area can be found using z-scores and standard normal tables or technology. The area to the left of a given z-score indicates the cumulative probability up to that point, which is essential for statistical inference and hypothesis testing.