Finding probability distribution of quantity depending on other distributions

Valentina Holland 2022-09-26 Answered
Finding probability distribution of quantity depending on other distributions
I 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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Answers (1)

Karli Moreno
Answered 2022-09-27 Author has 7 answers
Step 1
We 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 x
Step 2
If 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 )
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