Expectation for cos(x)^2

In summary, the question is about finding the average value of cos(x)^2 over a solid angle of a sphere which is 1/3. The correct integral to calculate is E[cos^2(x)] = 1/2 * ∫cos^2(x)sin(x)dx, with the correct weight being dΩ = d(cosθ)dψ and limits of integration from 0 to ∏ for θ and 0 to 2∏ for ψ.
  • #1
Zamze
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0

Homework Statement


This question comes from calculating the Einstein A and B coefficients. I am supposed to find the average value of cos(x)^2 over the solid angle of a sphere which is 1/3. And I need to show this.
A similar course in a different uni just says that For unpolarized, isotropic radiation, the expectation of cos(x)^2=1/3



Homework Equations



cos(2x)=2cos(x)^2-1



The Attempt at a Solution



I tried using the average integral equation however i always end up with 1/2. I've tried
1/pi *∫cos(X)^2dx and just use the trig equation that I have given. However the answer comes out as 1/2 and I do not know how to get 1/3. I also tried integrating from 0 to 2pi etc.

Thankful for any help!
 
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  • #2
You are getting wrong result because you don't weight the points on the sphere correctly. There are less points corresponding to each value of x near the poles. The correct uniform probability measure is [itex] \sin x dx [/itex], and the integral you should calculate is

[tex] E[\cos^2(x)] = \frac{1}{2} \int_0^\pi \cos^2(x) \sin(x) dx [/tex]
 
  • #3
clamtrox said:
You are getting wrong result because you don't weight the points on the sphere correctly. There are less points corresponding to each value of x near the poles. The correct uniform probability measure is [itex] \sin x dx [/itex], and the integral you should calculate is

[tex] E[\cos^2(x)] = \frac{1}{2} \int_0^\pi \cos^2(x) \sin(x) dx [/tex]

Ty very much.
 
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  • #4
In fact it is taken over solid angle dΩ and it is easy to write in spherical coordinates where
dΩ=d(cosθ)dψ and your average will be

=(∫(cos2θ) d(cosθ)dψ)/4∏,where 4∏ is ∫d(cosθ)dψ,limits are from o to ∏ for θ and 0 to 2∏ for ψ.
 
  • #5


I would approach this problem by first understanding the physical context in which the expectation for cos(x)^2 is being calculated. In this case, we are looking at the average value of cos(x)^2 over the solid angle of a sphere. This is related to the properties of unpolarized, isotropic radiation, which is a type of electromagnetic radiation that is equally likely to be emitted in any direction and has no preferred direction of polarization.

Next, I would look at the equations and definitions provided and try to understand how they relate to the problem at hand. The equation cos(2x)=2cos(x)^2-1 is useful in this case because it allows us to relate the expectation for cos(x)^2 to the expectation for cos(2x).

Using this equation, we can rewrite the expectation for cos(x)^2 as:

E[cos(x)^2]=1/2*E[1+cos(2x)]

Since the expectation for a constant value is just that value, we can simplify this to:

E[cos(x)^2]=1/2*(1+E[cos(2x)])

Now, we need to find the expectation for cos(2x) over the solid angle of a sphere. This can be done by integrating cos(2x) over the solid angle, which is equivalent to integrating over the full range of angles (0 to 2pi) in the x and y directions, and over the range of angles (0 to pi) in the z direction.

Integrating cos(2x) over these ranges gives us a value of zero, since cos(2x) is an odd function and the limits of integration are symmetric. Therefore, we can conclude that E[cos(2x)]=0.

Plugging this back into our equation for the expectation of cos(x)^2, we get:

E[cos(x)^2]=1/2*(1+0)=1/2

This is the same result you got when trying to use the average integral equation, but we have shown that it is not the correct answer for this problem.

To get the expected value of cos(x)^2 to be 1/3, we need to consider the fact that we are working with a sphere, which has a solid angle of 4pi steradians. Therefore, we need to divide our previous result by the total solid angle, giving
 

What is the expectation for cos(x)^2?

The expectation for cos(x)^2 is equal to 1/2, or 0.5. This means that on average, the value of cos(x)^2 will be 0.5.

How is the expectation for cos(x)^2 calculated?

The expectation for cos(x)^2 is calculated by taking the integral of cos(x)^2 from -∞ to ∞ and dividing it by the total range of the function. In this case, the total range is 2π, so the integral is divided by 2π.

What does the expectation for cos(x)^2 represent?

The expectation for cos(x)^2 represents the average value of the function cos(x)^2 over its entire range. It is a measure of the central tendency or typical value of the function.

Can the expectation for cos(x)^2 ever be negative?

No, the expectation for cos(x)^2 cannot be negative. This is because cos(x)^2 is always a positive function, and the expectation is calculated by taking the average of the function over its entire range.

How does the expectation for cos(x)^2 change with different values of x?

The expectation for cos(x)^2 does not change with different values of x. It is a constant value of 0.5, regardless of the value of x. However, the function cos(x)^2 itself will vary with different values of x.

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