which has determinant since (cos th sin thr it follows by the transformation rule ( that the joint pdf of and th is given by fr,th (rthfx, (xyr re th ( ) by integrating out th and rrespectivelywe find fr (rr - / and fth (th /( psince fr,th is the product of fr and fth the random variables and th are independent normal distribution multivariate normal distribution the normal (or gaussiandistribution -especially its multidimensional version -plays central role in data science and machine learning recall from table that random variable is said to have normal distribution with parameters and if its pdf is given by - ( (xe we write (us the parameters and are the expectation and variance of the distributionrespectively if and then (xe- / standard normal and the distribution is known as the standard normal distribution the cdf of the standard normal distribution is often denoted by ph and its pdf by ph in figure the pdf of the (us distribution for various and is plotted ( , / ( , ( , - - figure the pdf of the (us distribution for various and we next consider some important properties of the normal distribution theorem standardization let (us and define ( )/ then has standard normal distribution