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I am quoting this from DeGroot's "Probability and Statistics".

Let $X$ and $Y$ be random variables. The joint probability density function(p.d.f.) $f$ defines a surface over the $xy$-plane for which the height $f(x, y)$ at each point $(x, y)$ represents the relative likelihood of that point. For instance, if it is known that $Y = y_0$, then the point $(x, y)$ must lie on the line $y = y_0$ in the $xy$-plane, and the relative likelihood of any point $(x, y_0)$ on this line is $f(x, y_0)$.

I don't know the meaning of "relative likelihood".Please explain.

Silent
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I think the quote is using the term "likelihood" rather loosely. Likelihood is a separate concept from probability. What the joint density function gives at each point is a....density (hence the name). I don't know why the text said relative likelihood, which is $L_{relative}(a,b)=\frac{L(a)}{L(b)}$, where the likelihood function is evaluated given a set of data with the parameters becomming the variables: i.e., L(a|X) = P(X|a), where X is a set of data or outcomes and a is a parameter of the distribution.

The quote above appears to be talking about the conditonal probability of x given $y_{0}$, which is $p_{X|Y}(x|y_{0}) = p_{XY}(y_{0},x)/\int_x p_{XY}(y_{0},x) dx$, in that case the joint density evaluated at $y_{0}$ is acting in a similar manner to the likelihood function in Bayesian inference, but the two are rather different concepts.

Hope that helps.

  • Thanks for additional information. Actually, when I asked a question here, "relative probabilities" was mentioned for p.d.f. I am now confused!! – Silent Oct 16 '13 at 13:49
  • I took a look...I wouldn't use the term "relative probability" to describe a joint density, as relative implies you are comparing it relative to something else. I would compare probability density to concentration in chemistry...it is the mass (or in this case probability) you accumulate in a given interval of the random variables range. As an example, if you have 1 L of water with 1 g of salt in it, the total mass = 1 g, but depeneding on how well mixed the solution is, you can have local densities approaching pure salt in some parts of the bottle, which is > 1. –  Oct 16 '13 at 13:56
  • Many many hanks for taking troubles for me. – Silent Oct 16 '13 at 13:58
  • No problem. If it helped, you should select this question as answered so others know you don't need additional input. –  Oct 16 '13 at 14:37