What is the joint pdf for W=XY and Z=Y/X in the LaPlacian distribution?

In summary, the given joint pdf can be expressed as f(w,z)=(1/4)^2*4*[w*e^(w/2z)+z*w*e^(w/2z)] for 0<w,z<∞. The individual pdfs for X and Y can be obtained by integrating the given joint pdf over the appropriate limits. The use of Jacobian may not be applicable due to the presence of absolute values.
  • #1
marina87
22
0
A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity.

Find the joint pdf W=XY and Z=Y/X.

f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞
Someone told me I can not use Jacobian because of the absolute value. Is that true?
So far this is what I have but I feel like I am not going anywhere.

f(w,z)=(1/4)^2∫∫e^(-(|x|+|y|)/2)dxdy
=(1/4)^2∫∫[e^-|x|/2]*e^-|y|/2]
=(1/4)^2∫[e^-|x|/2]∫e^-|y|/2]

=(1/4)^2[∫[e^(-x/2)+∫e^(x/2))] * [∫[e^(-y/2)+∫e^(y/2))] the limits from -∞<x,y<0 and 0<x,y<∞

=[(1/4)^2 ]*4*[ ∫ [e^(x/2)dx] + ∫ [e^(y/2)dy] ] 0<x,y<∞
 
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  • #2
=[(1/4)^2]*4*[ x*e^(x/2) + y*e^(y/2) ] 0<x,y<∞ f(w,z)=(1/4)^2*4*[w*e^(w/2z)+z*w*e^(w/2z)] 0<w,z<∞
 

1. What is the LaPlacian joint distribution?

The LaPlacian joint distribution is a probability distribution that describes the relationship between two or more variables. It is a continuous distribution and is often used in statistical analysis and machine learning.

2. How is the LaPlacian joint distribution different from other joint distributions?

The LaPlacian joint distribution differs from other joint distributions, such as the Gaussian or Poisson distributions, in that it has a sharper peak at the mean and heavier tails. This makes it more suitable for modeling data with outliers or extreme values.

3. What are the main applications of the LaPlacian joint distribution?

The LaPlacian joint distribution is commonly used in image processing, signal processing, and machine learning applications. It is also used in Bayesian statistics as a prior distribution for Bayesian inference.

4. How is the LaPlacian joint distribution related to the LaPlacian operator?

The LaPlacian joint distribution is closely related to the LaPlacian operator, also known as the Laplace-Beltrami operator. This is because both are based on the concept of the Laplace equation, which describes the behavior of a physical system in terms of the distribution of potential energy.

5. What are the limitations of using the LaPlacian joint distribution?

One limitation of the LaPlacian joint distribution is that it assumes the variables are independent and identically distributed. In cases where this assumption does not hold, other joint distributions may be more appropriate. Additionally, the LaPlacian joint distribution may not be suitable for modeling data with a high degree of skewness or kurtosis.

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