The distribution of ratio of two uniform variables

In summary, the conversation discusses the calculation of the density of a ratio of two independent uniform random variables. The speaker mentions using a change of variables and a double integral, and then asks for clarification on their result not being a reasonable density. The second part of the conversation involves a question about computing the cumulative distribution function of a random variable using the joint cumulative distribution function of two independent random variables. The speaker mentions looking at the PDF of an exponential function and splitting it into two parts, but is unsure how to put everything together.
  • #1
gimmytang
20
0
Hello,
Let X ~ U(0,1), Y ~U(0,1), and independent from each other. To calculate the density of U=Y/X, let V=X, then:
[tex]f_{U,V}(u,v)=f_{X,Y}(v,uv)|v|[/tex] by change of variables.
Then:
[tex]f_{U}(u)=\int_{0}^{1}{f_{X,Y}(v,uv)|v|dv}=\int_{0}^{1}{vdv}={1\over 2}, 0<u<\infty[/tex], which is not integrated to 1.
Where I am wrong?
gim :cry:
 
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  • #2
you only integrated with respect to fu(u). Now you have to integrate with respect to fv(v). Or you could have just used a double integral to start with...
 
  • #3
Actually the marginal distribution of U, namely the distribution of the ratio of two uniform variables, is the only thing that I am interested. To be more clear:
[tex]f_{U}(u)={\int_{-\infty}^{\infty}f_{U,V}(u,v)dv}={\int_{0}^{1}f_{X,Y}(u,uv)|v|dv}={\int_{0}^{1}vdv}=1/2[/tex]
Now the question is my result 1/2 is not a reasonable density since it's not integrated to 1.
gim :bugeye:
 
  • #4
Question says Let X and Y be independent random variables with join cumulative distribution function (CDF) F subscript X,Y of (x,y)= P (X</= x, Y</=y). Show that the CDF Fz(z) of the random variable Z=min (X,Y) can be computed via
Fz(z)= Fx(z) + Fy(z) - Fx(z) . Fy(z) = 1 - (1 - Fx (z)) . (1- Fy (z))

Please reply to this asap. I need to submit this answer by Friday. Thanks!
 
  • #5
electroissues said:
Question says Let X and Y be independent random variables with join cumulative distribution function (CDF) F subscript X,Y of (x,y)= P (X</= x, Y</=y). Show that the CDF Fz(z) of the random variable Z=min (X,Y) can be computed via
Fz(z)= Fx(z) + Fy(z) - Fx(z) . Fy(z) = 1 - (1 - Fx (z)) . (1- Fy (z))

Please reply to this asap. I need to submit this answer by Friday. Thanks!
Don't jump into the thread of another. What have you done so far?
 
  • #6
Well, I'm new here and had problems starting a new thread.

I looked at PDF of an exponential function which is (1 - Fx (x)) and also since its also given its independent, we know it can be split into Fx (x) . Fy (y) but I can't put these things together.
 

What is the distribution of ratio of two uniform variables?

The distribution of ratio of two uniform variables follows a Beta distribution. This means that the resulting values will be between 0 and 1, and the distribution will be symmetric around the mid-point of 0.5.

Can the ratio of two uniform variables be used to model real-life situations?

Yes, the ratio of two uniform variables can be used to model real-life situations, such as the proportion of successes in a series of trials or the time it takes for a certain event to occur.

How does the shape of the distribution change when the two uniform variables have different ranges?

The shape of the distribution will remain the same, but the range of values will be scaled accordingly. For example, if one uniform variable has a range of 0-10 and the other has a range of 0-20, the resulting distribution will have a wider range of values between 0-1.

What happens to the distribution when one of the uniform variables is held constant?

When one of the uniform variables is held constant, the distribution will shift along the x-axis, but the shape of the distribution will remain the same. This means that the resulting values will still be between 0 and 1, but the range of values will change.

Can the distribution of ratio of two uniform variables be skewed?

No, the distribution of ratio of two uniform variables will always be symmetric around the mid-point of 0.5. It cannot be skewed to one side or the other.

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