Thread: Cauchy distribuation and maths help

Hello i have this question that i am struggling to answer regarding changing variables, i really dont know how to do it

A gun fires at random in the angular range −π/2 < θ < π/2 towards a wall a distance 'l' away. If 'y' is the coordinate along the wall,


show that

g(y)dy=(1/π) (1/(1 +(y/l)²)) (dy/l)










This is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L either side of the peak. Calculate the new normalisation constant, and find the standard deviation.

id appreciate if someone could even point me in the right direction, im abit stumped...

Originally Posted by bobby84

Hello i have this question that i am struggling to answer regarding changing variables, i really dont know how to do it

A gun fires at random in the angular range −π/2 < θ < π/2 towards a wall a distance 'l' away. If 'y' is the coordinate along the wall,



show that

g(y)dy=(1/π) (1/(1 +(y/l)²)) (dy/l)










This is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L either side of the peak. Calculate the new normalisation constant, and find the standard deviation.

I am not 100% sure what you are asking, but I think this is what you mean....





From the diagram we see that



If we take the derivative we get



We can rewrite the secant function interms of tangent using the pythagorean identity



Now substitute tangent out for y to get



Solving for gives



So for this to be a distribution the integral over the entire real line must be 1. You can use this to find the constant.

im not sure if you read this bit...

This is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L either side of the peak. Calculate the new normalisation constant, and find the standard deviation.

Originally Posted by bobby84

im not sure if you read this bit...

This is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L either side of the peak. Calculate the new normalisation constant, and find the standard deviation.

Well if

Then we need to calculate the integral



So now try to calculate this integral. What do you get?

hey i got there in the end actually,

my final solution was....

(sigma) = sqrt(4-(pie)/2(pie)) l

cheers for the help