What's the PDF of a Unit Sphere Projection on the X-Axis?

AI Thread Summary
To find the probability density function (pdf) of the projection of a unit random vector on the x-axis, one must consider the uniform distribution of points on the surface of a unit sphere. The probability of projecting onto a specific area on the x-axis is proportional to the area of the corresponding spherical cap divided by the total surface area of the sphere. This relationship indicates that the pdf is derived from the geometry of the sphere and the area of the projection. The discussion emphasizes the importance of understanding the area-to-total surface area ratio for calculating the probability of the projection. The solution involves integrating over the relevant areas to obtain the final pdf.
Micle
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Please help, I have an assignment's question but I don't idea how to work on it. The question is that:

Let R be a unit random vector points on the surface of a unit sphere. If the probabilty of R is uniform over the entire surface of a unit sphere, find the pdf of the projection of R on the x-axis

any idea to do so?
 
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Hi Micle

so in this example area/total sphere area is probability to find your vector within the given area

so the probability of having a projection will be related to the area area/total sphere area for a given dx incremnent at that position
 
Hi Lanedance

Thanks!
 
cheers
 

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