Density of transformed random variables

In summary, the conversation discusses the use of transformations of random variables and the Jacobian determinant to find the density of a transformed random variable. The solution involves creating a new variable and using the change of variable theorem to find the density. There are questions about the choice of variables and their relation to regions and the change of variables theorem. It is also mentioned that it is possible to use different mappings and probability spaces for the transformed variables.
  • #1
RandomVariabl
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I'm studying for the probability actuarial exam and I came across a problem involving transformations of random variable and use of the Jacobian determinant to find the density of transformed random variable, and I was confused about the general method of finding these new densities. I know the Jacobian matrix is important in the change of variable theorem, but I am having trouble connecting the vector calculus concepts with probability distributions.

Let X, Y have a joint pdf of f(x,y) = e^(-x-y), x>0, y>0. Find the density of U = e^(-x-y).
Solution: The solution creates a new variable V=X=h(u,v), and let's Y= -lnU-V = k(U,V), and finds a new joint pdf g(u,v)=f(V,-lnu-v)*J[h,k] = 1,where J[h,k] is the Jacobian determinant of h, and k. Then, fu(u) = integral(0,-ln u) g(u,v)dv = -ln U, as 0<V<-ln U.

*I am confused why we let V = X...I figure we can we also let V = Y from symmetry and get the same result, but can we let V = 2X, or V= X^2, or V=r(X),r arbitrary function(would r need special properties such as being injective, or bijective?), when computing the density fu. Do

How can we map (x,y) to (u,v)? From my limited understanding, by creating a new variable V, we are mapping the region of positive R^2 to some other region containing of which (u,v) come from. Can anything be said about these regions? Am I missing some fundamental assumptions? How does this relate to the change of variables theorem and its application in finding fu? I am interested in the vector analysis of this problem. Is it possible that there are several mappings with varying probability spaces of (u,v) and varying g(u,v)'s, and integral(limits of v) g(u,v)dv = fu(u), the one distinct density of U (this is what's is what seems strange and fuzzy to me)?

I apologize if any of my questions are not clear. If this is the case, try to clarify the stuff near *.
 
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  • #2
Well yes you could use any invertible transformation you like as long as you're careful to define the domain and range and inverse transformation.

Personally I prefer to avoid densities wherever possible and find the CDF instead. For this example it's not hard to show that P[U<=u]=u(1-log(u)) for 0<=u<=1, then differentiate to get the pdf if you must.
 

Related to Density of transformed random variables

What is the definition of density of transformed random variables?

The density of transformed random variables is a statistical concept that measures the probability of a random variable taking on a specific value or falling within a certain range of values. It is an important tool in probability theory and is used to analyze and describe the behavior of random variables.

How is the density of transformed random variables calculated?

The density of transformed random variables can be calculated using the transformation rule, which involves taking the derivative of the inverse transformation function. This allows us to determine how the probability of the transformed variable changes with respect to the original variable.

What are some common examples of transformed random variables?

Some common examples of transformed random variables include the log-normal distribution, which is a transformation of the normal distribution, and the chi-squared distribution, which is a transformation of the normal distribution squared. Other examples include the exponential and gamma distributions, which are both transformations of the normal distribution.

What is the relationship between the density of transformed random variables and the original random variables?

The density of transformed random variables is related to the original random variables through the transformation function. This function maps the original values onto the transformed values, and the density of the transformed variable is calculated using the original density and the transformation function.

Why is the concept of density of transformed random variables important in scientific research?

The density of transformed random variables is important in scientific research because it allows us to model and analyze complex systems that involve multiple variables. It also helps us to better understand the relationships between different variables and their impact on the overall system. Additionally, it is a key tool in statistical analysis and hypothesis testing.

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