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The standard normal distribution is a special type of normal distribution. It represents a bell-shaped curve where the mean (\(\mu = 0\) ) and the standard deviation (\(\sigma = 1\) ) are both fixed at particular values.
The random variable \(Z\) , which follows a standard normal distribution, helps in simplifying and standardizing problems involving normal distributions.
Key properties include:

• Mean of 0: It indicates that the distribution is centered around zero.
• Standard deviation of 1: This measures the spread of the distribution.
• The total area under the curve equals 1, showing 100% of the probabilities are covered within the curve.

The standard normal distribution is central in probability and statistics, as it provides the basis for the standardization of other normal distributions. By converting a normal variable into a standard normal variable, it becomes easier to compute probabilities and percentiles across different datasets.

The probability density function (PDF) of a distribution provides crucial information about how the probabilities of a continuous random variable are distributed.
For continuous distributions like the normal distribution, the PDF helps determine the likelihood a random variable falls within a particular range of values.
In the context of the standard normal distribution, the PDF is denoted as \(\phi(z)\).
This function describes the shape of the standard normal curve, which has its highest point at the mean zero and rolls off symmetrically.
To get the PDF of a normal distribution with different mean \(\mu\) and standard deviation \(\sigma\), a transformation of the form \(Y = \sigma Z + \mu\) is applied. This modifies the PDF to \(\phi\left(\frac{y - \mu}{\sigma}\right)\cdot \frac{1}{\sigma}\), which accounts for the shifted and scaled version of the original distribution. Key Points:

• The PDF is vital for determining risk and probabilities in various applications, such as finance and science.
• The shape of the PDF affects the probability calculations and the spread of possible outcomes.
• The area under the PDF between two points represents the probability of the variable falling within that interval.

The cumulative distribution function (CDF) gives a complete understanding of the probabilities of a random variable being less than or equal to a certain value in a distribution.
For the standard normal distribution, the CDF is denoted as \(\Phi(z)\) . It describes the total probability from the leftmost end of the distribution up to a specific point, effectively providing the probability that the variable takes on a value less than or equal to a particular number.
Main Characteristics:

• A CDF starts from 0 and asymptotically approaches 1, reflecting the total probability.
• In transformations involving a normal distribution, such as \(Y = \sigma Z + \mu\), the CDF helps determine how values are distributed based on the new mean and standard deviation.
• The CDF is particularly useful when finding percentiles or quartiles within a distribution.

When dealing with different normals, understanding the CDF helps relate to probabilities about outcomes concerning mean and variance, making it invaluable in statistics and the study of variations across datasets.