Parameters of a Distribution

When referring to a Probability Distribution, they have unique characteristics. These characteristics can be referred to as a Parameters.

Different Parameters:

• Mean ($\mu$): The average value of the distribution. For a normal distribution, it determines the center of the distribution.
• Variance (${\sigma }^2$): Measures the spread of the distribution; how far the values are from the mean. In a normal distribution, it determines the width of the bell curve.
• Standard Deviation ($\sigma$): The square root of the variance, also a measure of spread.
• Probability ($p$): In a binomial distribution, $p$ is the probability of success in a single trial.
• Number of Trials ($n$): In a binomial distribution, $n$ is the number of independent trials.
• Rate ($\lambda$): In a Poisson distribution, $\lambda$ is the average number of events in a given interval.
• Shape Parameters ($\alpha$, $\beta$, etc.): Some distributions, like the beta or gamma distributions, have shape parameters that determine the skewness and kurtosis of the distribution.

Symbols for it

$\theta$ represents a generic parameter and $\hat{\theta }$ for it’s estimator. E.G. $\hat{\mu }=\bar{X}$ means “the point estimator of the population mean $\mu$ is the sample mean $\bar{X}$”