Normal and Exponential Distribution

Normal Distribution

The Normal Distribution is a continuous probability distribution that is symmetrical and bell-shaped, describing data that clusters around a mean (average). It is also known as the Gaussian distribution.

Formula:

The probability density function (PDF) of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is given by:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$

where:

$\mu$ is the mean (average) of the distribution

$\sigma$ is the standard deviation of the distribution

Example:

Consider the heights of adult males in a population. If the heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches, the PDF of the height X can be written as:

$f(x) = \frac{1}{3 \sqrt{2\pi}} e^{-\frac{(x - 70)^2}{2 \cdot 3^2}}$

This function describes the probability density of different heights around the mean height of 70 inches.

Characteristics:

The normal distribution is defined by two key parameters:

$\mu$: the mean, which determines the center of the distribution

$\sigma$: the standard deviation, which measures the spread or dispersion of the distribution

The mean (expected value) of a normal distribution is $\mu$

The variance of a normal distribution is $\sigma^2$

Approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Exponential Distribution

The Exponential Distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is often used to model the time until the next event, such as the time between arrivals of customers at a service point.

Formula:

The probability density function (PDF) of an exponential distribution with rate parameter $\lambda$ is given by:

$f(x) = \lambda e^{-\lambda x}$

where:

$\lambda$ is the rate parameter, which is the inverse of the mean (i.e., $\lambda = \frac{1}{\mu}$)

x is the time between events

Example:

Suppose the average time between arrivals of buses at a bus stop is 10 minutes. The rate parameter $\lambda$ is the reciprocal of the mean, so $\lambda = \frac{1}{10} = 0.1$ per minute. The PDF of the time between bus arrivals X is:

$f(x) = 0.1 e^{-0.1 x}$

This function describes the probability density of different times between bus arrivals.

Characteristics:

The exponential distribution has one key parameter:

$\lambda$: the rate parameter, which is the inverse of the mean

The mean (expected value) of an exponential distribution is $\mu = \frac{1}{\lambda}$

The variance of an exponential distribution is $\sigma^2 = \frac{1}{\lambda^2}$

The exponential distribution is memoryless, meaning that the probability of an event occurring in the next interval is independent of how much time has already elapsed.