Define standard normal distribution.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

The standard normal distribution is a probability distribution that is widely used in statistics and probability theory. It is a normal distribution with a mean of 0 and a standard deviation of 1. The probability density function of the standard normal distribution is given by:

f(x) = (1/√(2π))e^(-x^2/2)

where x is a random variable, e is the base of the natural logarithm, and π is the mathematical constant pi.

The cumulative distribution function of the standard normal distribution is denoted by Φ(x) and is given by:

Φ(x) = (1/√(2π)) ∫(-∞,x) e^(-t^2/2) dt

where ∫(-∞,x) denotes the integral from negative infinity to x.

The standard normal distribution is important because it can be used to standardize any normal distribution. This means that any normal distribution can be transformed into a standard normal distribution by subtracting the mean and dividing by the standard deviation. This process is called standardization and is given by:

z = (x - μ) / σ

where z is the standardized variable, x is the original variable, μ is the mean of the original variable, and σ is the standard deviation of the original variable.

In summary, the standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is important because it can be used to standardize any normal distribution, which is useful in statistical analysis.