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Let us say, we have two random variables, x1 --> U(10,20) i.e. x1 is uniformly distributed between 10 and 20, and x2 --> U(20,40) i.e. x2 is uniformly distributed between 20 and 40.

Moreover, it has to be ensured that x1+x2 = 40 always.

How do I compute the joint distribution P(X1,X2|X1+X2=40)?

In other words, how do I find joint distribution based on a sum constraint?

user11151416
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1 Answers1

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The joint density function of continuous random variables $X$ and $Y$ conditioned on $X+Y=a$ does not exist. Conditioned on $X+Y=a$, the random point $(X.Y)$ is constrained to lie on the straight line through $(a,0)$ and $(0,a)$ and so doesn't have a joint density in the sense of $f_{X,Y}(x,y)$ telling you how dense the probability mass is in the vicinity of $(x,y)$: the units are probability mass per unit area and since $(X,Y)$ is constrained to lie on a line (which has zero area!), the only nonzero "value" we can ascribe to the joint density is $\infty$. What we do have is a line density measured in probability mass per unit length.

Dilip Sarwate
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  • I really want to thank you for answering my question, I really really appreciate it. So basically, what happens is if we are given n measurements each of which are uniformly distributed within certain intervals, and that the total sum of the n measurement is always constant, the joint density always loses a degree of freedom.

    If n=3, then instead of evaluating probabilities on volume, it is evaluated on surface. Could you please give an example of how to actually compute it? I tried by writing out the joint pdf of two uniform densities but was unable to accomodate the sum constraint

     – user11151416 Apr 01 '19 at 21:30