Finding probability distribution of quantity depending on other distributionsI have a vector that depends on the coordinates of randomly drawn unit vectors in R 2 : σ = ( cos ⁡ ( θ a ) + cos ⁡ ( θ b ) 1 + cos ⁡ ( θ a ) cos ⁡ ( θ b ) sin ⁡ ( θ a ) + sin ⁡ ( θ b ) 1 + sin ⁡ ( θ a ) sin ⁡ ( θ b ) ) Here, θ a , θ b ∈ [ 0 , 2 π ), are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in R 2 . I am interested in figuring out a probability distribution for the coordinates of the σ-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.My questions are thus:Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?If the method exists, how would I use it on my concrete example?

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Finding probability distribution of quantity depending on other distributionsI have a vector that depends on the coordinates of randomly drawn unit vectors in             R        2  :  σ  =      (                                                      cos              ⁡              (                              θ                a                            )              +              cos              ⁡              (                              θ                b                            )                                      1              +              cos              ⁡              (                              θ                a                            )              cos              ⁡              (                              θ                b                            )                                                                                      sin              ⁡              (                              θ                a                            )              +              sin              ⁡              (                              θ                b                            )                                      1              +              sin              ⁡              (                              θ                a                            )              sin              ⁡              (                              θ                b                            )                                            )  Here,       θ    a    ,      θ    b    ∈  [  0  ,  2  π  ), are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in             R        2  . I am interested in figuring out a probability distribution for the coordinates of the   σ-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.My questions are thus:Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?If the method exists, how would I use it on my concrete example?

Karli Moreno · Accepted Answer

Step 1We know that the PDF (probability density function) of the random variable   θ is   P  D  F  (  θ  )  =      lim          Δ      x      →      0                  P      (      θ      ∈      (      x      ,      x      +      Δ      x      )      )              Δ      x        →  P  (  θ  ∈  (  x  ,  x  +  Δ  x  )  )  =  P  D  F  (  θ  )  Δ  xStep 2If the random variable   σ is dependent of   θ by   y  =  y  (  x  ) we must have  P  (  θ  ∈  (  x  ,  x  +  Δ  x  )  )  =  P  (  σ  ∈  (  y  ,  y  +  Δ  y  )  )  →  P  D  F  (  θ  )  Δ  x  =  P  D  F  (  σ  )  Δ  y  →  P  D  F  (  σ  )  =  l  i      m          Δ      x      →      0                  Δ      x              Δ      y        P  D  F  (  θ  )  =            d      x              d      y        P  D  F  (  θ  )

Finding probability distribution of quantity depending on other distributions

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