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