I am new to multivariate analysis and without a great math background I have been able to follow the book Applied multivariate statistical analysis up until this section...

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The way I am able to follow the book is mostly through the graphical aids; the math part is not easy to understand for me... I picture bivariate normal, as weight and height of (ie 1000 people)... I can picture in my head the density (with clear correlation)...

My question is: per the book if X1 or X2 is given, the density changes... how come the density changes? When I picture height and weight as a bivariate density, there is no way the density could change, I see both height and weight as given.

(I do understand the 'given' concept in probability, but apparently not here...)


1 Answer 1


I was going to plot it myself, but looking online, this page does such a good job of it already. They even let you zoom in and swivel the graph. Look at the 2nd from the last graph, the 3d one. So if p(y|x=4), first go to x=4 on the x axis and then look across all the values of y, the z axis now tells you the probability density of y given x=4. Across all the y's, what is the range of the z axis (the density of the probability), not very large.

Looks like they get a maximum probability density near y=0 of approximately 0.005 at x=4. That is, p(y=0|x=4)=0.005 (approximately).

Now look at it when x=0. now the probabilities for y are much higher. p(y=0|x=0) is around 0.03.

Without being conditional on x, y is still a normal distribution, that is, it'll still be that bell shaped curve, but the precise nature of that bell shaped curve changes a lot once you condition it on x. In graphical terms, in this example, it flattens y's curve out if the x value is far away from 0 (in the tails).

  • $\begingroup$ So, with my height, weight example would be, a function of height, given weight... For example (height|weight=80kg)? And this would still be normal? $\endgroup$ Dec 18, 2019 at 15:52
  • $\begingroup$ Yes, f(X_1|X_2) is normal if X_1 and X_2 are normal. For other combinations of distributions, you use the formula above in order to calculate that particular conditional distribution. $\endgroup$
    – Huy Pham
    Dec 18, 2019 at 16:12
  • $\begingroup$ Just to help clarify more (hopefully it clarifies instead of confuses). The function f tells you the probability of any y given any x. You could sub in any particular x value then it would tell you the probability of any y given that particular x. Subbing in a particular y as well, then tells you the probability of that y given that particular x. Sub in y without subbing in x and it'll still be a function; a function that describes the prob of y in terms of possible x values (i.e. given x) $\endgroup$
    – Huy Pham
    Dec 18, 2019 at 16:17

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