Finding Density: Method to Tell Difference

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In summary, the marginal density of Y is the density of Y when you ignore X, and to do that, you must integrate out X.
  • #1
semidevil
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These are 2 problems that I can solve, but I don't know how to tell the difference between these 2 when it comes to finding the densities. I just need to know how to tell whichmethod to use to find the density. I do not need the solution.

Homework Statement



A)
Suppose that (X; Y ) is uniformly chosen from the set given by 0 < X < 3 and x < y < root(3x). Find the marginal density fy (y) of Y.

B)
If X is uniformly distributed on [0, 2], and given that X = x, Y is uniformly distributed on [x 2x], what is P[Y <2]?

2. The attempt at a solution
For A), to find the joint density, I integrate 1 dy dx of the shape to get the area. the joint density is then 1/area. This makes sense to me.
For B), integrating 1 dydx doesn't seem to work and instead, it is just simply combining f(x) and f(y). 1/2 * 1/x to get the density.

I understand the uniform shortcuts so I know where 1/2 and 1/x came from, but how do I know when to use which method? I,e. how do I know that I need to integrate 1, rather then just multiply f(x)*f(y).
both tell us that x, y are uniformly distributed; I understand that the difference is that one involves a conditional distribution, so is that the determining factor?3. Relevant equations
if f(x) is uniform on (a,b), the area is b-a. the density would then be 1/(b-a).
f(x)*f(y) = f(x,y) if independent.
 
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  • #2
semidevil said:
These are 2 problems that I can solve, but I don't know how to tell the difference between these 2 when it comes to finding the densities. I just need to know how to tell whichmethod to use to find the density. I do not need the solution.

Homework Statement



A)
Suppose that (X; Y ) is uniformly chosen from the set given by 0 < X < 3 and x < y < root(3x). Find the marginal density fy (y) of Y.

B)
If X is uniformly distributed on [0, 2], and given that X = x, Y is uniformly distributed on [x 2x], what is P[Y <2]?

2. The attempt at a solution
For A), to find the joint density, I integrate 1 dy dx of the shape to get the area. the joint density is then 1/area. This makes sense to me.
For B), integrating 1 dydx doesn't seem to work and instead, it is just simply combining f(x) and f(y). 1/2 * 1/x to get the density.

I understand the uniform shortcuts so I know where 1/2 and 1/x came from, but how do I know when to use which method? I,e. how do I know that I need to integrate 1, rather then just multiply f(x)*f(y).
both tell us that x, y are uniformly distributed; I understand that the difference is that one involves a conditional distribution, so is that the determining factor?


3. Relevant equations
if f(x) is uniform on (a,b), the area is b-a. the density would then be 1/(b-a).
f(x)*f(y) = f(x,y) if independent.

Neither of your X and Y are independent, so that last equation is useless.

Think of what density really means:
[tex] P(x < X < x + \Delta x, y < Y < y + \Delta y) \doteq f(x,y) \Delta x \, \Delta y [/tex]
(neglecting smaller-order terms like ##(\Delta x)^2,## etc).
This implies
[tex] P(y < Y < y+\Delta y | X=x)
\equiv \lim_{\Delta x \to 0} P(y < Y < y+\Delta y|x < X < x + \Delta x),[/tex]
and this last conditional probability is
[tex]P(y < Y < y+\Delta y|x < X < x + \Delta x) =
\frac{P(x < X < x + \Delta x, y < Y < y + \Delta y)}{P(x <X < x + \Delta x)}[/tex]
In other words, for small ##\Delta x, \, \Delta y## we have
[tex] P(x < X < x + \Delta x, y < Y < y + \Delta y) \doteq
P(y < Y < y+\Delta y | X=x) \cdot P(x <X < x + \Delta x)[/tex]
You can take it from here.
 

Related to Finding Density: Method to Tell Difference

1. What is density?

Density is a measure of how much mass is contained in a given volume of a substance. It is commonly expressed in units of grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

2. Why is it important to find density?

Density is important because it can help identify and distinguish between different substances. It is also a physical property that can provide useful information about a material's composition and behavior under certain conditions.

3. What is the formula for calculating density?

The formula for calculating density is: density = mass / volume. This means that to find the density of a substance, you need to know its mass and volume.

4. How do you measure density?

To measure density, you need to first find the mass of the substance using a balance or scale. Then, you need to measure the volume of the substance using a graduated cylinder or by calculating the volume of a regular-shaped object. Finally, divide the mass by the volume to calculate the density.

5. How can density be used to differentiate between substances?

Density can be used to differentiate between substances because each substance has a unique density. If two substances have different densities, they will also have different masses and volumes for the same amount of material. This can help identify unknown substances or determine if a sample is pure or a mixture.

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