Univariate and Bivariate Normal Distributions 

The following was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com 

1) A Univariate Normal Distribution 

A univariate normal distribution has a probability density function equal to 

(1)

 

with mean  

(2)

 

and standard deviation  

(3)

 

We can plot a univariate normal distribution as follows 

We should also note that changing the values of the mean and standard deviation results in different shapes and of 

the Probability Density Function (PDF) as seen in the figures below where normal [mean, standard deviation]. 

Different values for the mean give us: 

Different values for the standard deviation 

A standard normal distribution has a probability density function equal to 

(4)

 

with mean  

(5)

 

and standard deviation  

(6)

 

We can plot a standard normal distribution as follows 

We should also note that a standard normal distribution has an kurtosis equal to 3. 

(7)

 

2) A Bivariate Normal Distribution 

A bivariate normal distribution has a probability density function equal to 

(8)

 

We can plot the bivariate normal distribution if we assume different values of  

We can now simulate how the bivariate normal distribution would look with different mean values of x (just click on the chart and click play) 

We can now simulate how the distribution would look with different mean values of y (just click on the chart and click play) 

We can now simulate how the bivariate normal distribution would look with different standard deviation values of x  

(just click on the chart and click play) 

We can now simulate how the bivariate normal distribution would look with different standard deviation values of y  

(just click on the chart and click play) 

For a standard normal distribution (mean=0 and standrad deviation=1) so we get:
 

(9)

 

We can plot the standard normal distribution if we for example assume that the correlation coefficient ρ is 0 

We can now simulate how the distribution would look with different values of (just click on the chart and click play) 

We can now plot the all distributions for for different values of  

We can also plot the contour plots 

The End ! 

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