# Question

Suppose we are given samples of the CDF of a random variable. That is, we are given Fn = Fx (xn) at several points, xn Ɛ { x1, x2, x3,….xk. After examining a plot of the samples of the CDF, we determine that it appears to follow the functional form of a Rayleigh CDF,

The object of this problem is to determine what value of the parameter, σ2, in the Rayleigh CDF best fits the given data.

(a) Define the error between the th sample point and the model to be

Find an equation that the parameter σ2 must satisfy if it is to minimize the sum of the squared errors,

You probably will not be able to solve this equation, you just need to set up the equation.

(b) Next, we will show that the optimization works out to be analytically simpler if we do it in the log domain and if we work with the complement of the CDF. That is, suppose we redefine the error between the th sample point and the model to be

Find an equation that the parameter σ2 must satisfy if it is to minimize the sum of the squared errors. In this case, you should be able to solve the equation and find an expression for the optimum value of σ2.

The object of this problem is to determine what value of the parameter, σ2, in the Rayleigh CDF best fits the given data.

(a) Define the error between the th sample point and the model to be

Find an equation that the parameter σ2 must satisfy if it is to minimize the sum of the squared errors,

You probably will not be able to solve this equation, you just need to set up the equation.

(b) Next, we will show that the optimization works out to be analytically simpler if we do it in the log domain and if we work with the complement of the CDF. That is, suppose we redefine the error between the th sample point and the model to be

Find an equation that the parameter σ2 must satisfy if it is to minimize the sum of the squared errors. In this case, you should be able to solve the equation and find an expression for the optimum value of σ2.

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