- #1
Master1022
- 611
- 117
- Homework Statement
- If we have ## p(x) = c(x+1)^2 (1 - x) ## where ## -1 \leq x \leq 1 ## and we have a variable transformation ## Z = X^2 ##, then find ## p(z) ##
- Relevant Equations
- Probability transformation
Hi,
I have a question about probability transformations when the transformation function is a many-to-one function over the defined domain.
Question: How do we transform the variables when the transformation function is not a one-to-one function over the domain defined? If we have ## p(x) = m(x+1)^2 (1 - x) ## where ## -1 \leq x \leq 1 ##, where ## m ## is a constant, and we have a variable transformation ## Z = X^2 ##, then find ## p(z) ##
Context attempt:
I was reading some lecture notes where all it says is: "if the transformation function ## h(x) ## is not one-to-one, then we use a more complicated method".
So I know that usually when we go from ## x ## to ## z ##, then we need to consider the Jacobian determinant ## | \frac{\partial x}{\partial z} | ##. For the example above, then that becomes:
[tex] p(z) = m(\sqrt{z} + 1)^2 (1 - \sqrt{z}) \cdot \left|\frac{\partial x}{\partial z} \right| = m(\sqrt{z} + 1)^2 (1 - \sqrt{z}) \cdot \frac{1}{2\sqrt{z}} [/tex]
but this exactly what was done for the situation where the transformation function was 1-to-1 over the domain defined.
For a discrete system when we have a transformation, then we might have something along the lines of:
[tex] p(Z = z) = \sum p(Z = X^2, X = x) = \sum p(Z = X^2|X = x) p(X = x) [/tex]
but I am confused on how I can utilize this methodology for the continuous case.
Thank you in advance for any help.
I have a question about probability transformations when the transformation function is a many-to-one function over the defined domain.
Question: How do we transform the variables when the transformation function is not a one-to-one function over the domain defined? If we have ## p(x) = m(x+1)^2 (1 - x) ## where ## -1 \leq x \leq 1 ##, where ## m ## is a constant, and we have a variable transformation ## Z = X^2 ##, then find ## p(z) ##
Context attempt:
I was reading some lecture notes where all it says is: "if the transformation function ## h(x) ## is not one-to-one, then we use a more complicated method".
So I know that usually when we go from ## x ## to ## z ##, then we need to consider the Jacobian determinant ## | \frac{\partial x}{\partial z} | ##. For the example above, then that becomes:
[tex] p(z) = m(\sqrt{z} + 1)^2 (1 - \sqrt{z}) \cdot \left|\frac{\partial x}{\partial z} \right| = m(\sqrt{z} + 1)^2 (1 - \sqrt{z}) \cdot \frac{1}{2\sqrt{z}} [/tex]
but this exactly what was done for the situation where the transformation function was 1-to-1 over the domain defined.
For a discrete system when we have a transformation, then we might have something along the lines of:
[tex] p(Z = z) = \sum p(Z = X^2, X = x) = \sum p(Z = X^2|X = x) p(X = x) [/tex]
but I am confused on how I can utilize this methodology for the continuous case.
Thank you in advance for any help.