Statistics

A two-dimensional random variable with a uniform distribution

A two-dimensional random variable with a uniform distribution

A two-dimensional random variable with a uniform distribution

A two-dimensional random variable with a uniform distribution

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 2Marginal probability density function - variable 1

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

variable 1

variable 2Marginal probability density function - variable 1

mean

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

variable 1

variable 2Marginal 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 variablewe need…..

A single random variable with a multi-normal distribution

mean

standard deviation

µ

mean value variance

σ2

To characterise two random variableswe need…..

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

variable 1

variable 2

µ1 µ2

mean valuesvariance-covariancematrix

σ12 σ12

σ21 σ22

µ1 µ2

mean valuesvariance-covariancematrix

σ12 σ12

σ21 σ22

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

variable 1

variable 2Marginal 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

µ1 µ2

mean valuesvariance-covariancematrix

σ12 σ12

σ21 σ22

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

µ1 µ2

mean valuesvariance-covariancematrix

σ12 0

0 σ22