How to generate conditional probability from a algebric equation ?

[viperkill]

How can I find the conditional probability density function of a dependent variable given the independent variable set. Say, Y is some deterministic function of a set of variables X , or Y=f(X)+e.

How can I fine the conditional pdf or P(Y|X) ?

 

[statdad]

If $$\mathbf{X}$$ a set of constants? Then, assuming $$e$$ has distribution function $$G$$,

$$H(y) \equiv \Pr(Y \le y) = \Pr(e \le y - f(\mathbf{X})) = G(y-f(\mathbf{X})$$

If $$G$$ has a density then

$$h(y) = g(y-f(\mathbf{X}))$$

If $$\mathbf{X}$$ is random with joint distribution function $$W$$, as long as they are independent of $$e$$, you can argue this way. If the $$\mathbf{X}$$
are given (fixed), then you are in the case discussed above, and

$$H(y \mid \mathbf{X}) = G(y - \mathbf{X}), \quad h(y-\mathbf{X}) = h(y- \mathbf{X})$$

 

[viperkill]

Thank you for your reply. Can you exemplify your explanation.

Suppose, y=0.25x₁+5.32x₂+0.356x₃+e.
where e is normally distributed or poisson distributed.
What will be the pdf of Y|X ?

if X has some probability distribution, then what will be the solution?

 

[Stephen Tashi]

if X has some probability distribution, then what will be the solution?

Whoa! I thought you said the $$X_i$$ were constants. Are they now random variables?

 

[viperkill]

if X were ordinary variables then what will be the solutions and what if X are random variable ?

 

[Stephen Tashi]

If you want specific answers, you'll have to give specific information. For example, are the $$X_i$$ mutually independent? Are they identically distributed?

Are you familiar with convolutions of distributions?

 

[viperkill]

If Xi are ordinary variables, like y=0.25x1+5.32x2+0.356x3+e. then what will be the solution ??

if Xi are mutually independent and identically distributed (iid) random variables. Then what will be the P(Y|X) for both cases ?

 

[Stephen Tashi]

Let's say that e is independent of the $$X_i$$ and has density $$f(e)$$. I'll interpret $$P(y|x)$$ to mean the density of $$y$$. For given numerical values of the $$X_i$$, the density of $$y$$ is $$f( y - (0.25 x_1 + 5.32 x_2 + 0.356 x_3) )$$.

A density $$f(e)$$ is sometimes given by formula that only applies on some subset of the real numbers (e.g. the poission). It is understood that on numbers outside of this subset, the density is defined to be zero So if you are writing a computer program, you should test whether the value y - (0.25 x_1 + 5.32 x_2 + 0.356 x_3) is in the subset where the formula applies.