Statistics

A two-dimensional random variable with a uniform distribution

variable 1 variable 2

Probability density function for variable 1 value probability

A two-dimensional random variable with a uniform distribution variable 1 variable 2

Probability density function for variable 2 value probability

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 Marginal probability density function - variable 1

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 Marginal probability density function - variable 1 mean

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 Marginal probability density function - variable 1 mean standard deviation

A single random variable with a multi-normal distribution mean standard deviation

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 marginal probability density function - variable 2 mean standard deviation

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 conditional probability density function - variable 1

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 conditional probability density function - variable 1

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 conditional probability density function - variable 1

Two-dimensional probability density function

To characterise a single random variable we need…..

A single random variable with a multi-normal distribution mean standard deviation

µ mean value variance σ2σ2

To characterise two random variables we need…..

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2

µ 1 µ 2 mean values variance-covariance matrix σ 1 2 σ 12 σ 21 σ 2 2

µ 1 µ 2 mean values variance-covariance matrix σ 1 2 σ 12 σ 21 σ 2 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 Marginal probability density function - variable 1 mean standard deviation

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 marginal probability density function - variable 2 mean standard deviation

µ 1 µ 2 mean values variance-covariance matrix σ 1 2 σ 12 σ 21 σ 2 2

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 Parameter correlation

A two-dimensional random variable with a multi-normal distribution variable 1 variable 2 No parameter correlation

µ 1 µ 2 mean values variance-covariance matrix σ σ 2 2