Minimized Risk (Probability Theory)

[freeski]

Homework Statement

The rv X has pdf f(x) = cx, for 1 < x < 2, with f(x) = 0 otherwise. Compute the 3 M’s, and also compute the minimized risks in each case. Mode, median, mean.

Homework Equations

The Attempt at a Solution

I think I have computed the 3 M's
mode of x = 2
median of x = 1.225
mean of x = 1.56

I am not sure what is meant by minimized risk. I have been going over my notes and all I have are the derivations of the 3 M's.

Thank you.

 

[mighty2000]

Thank you for your post. It looks like you have correctly computed the mode, median, and mean for the given probability density function (pdf). As for the minimized risk, this refers to the minimum value of the risk function for this particular probability distribution.

The risk function is a measure of the potential loss or error associated with a decision or action. In this case, the decision or action would be choosing a particular value for the random variable X. The risk function is given by:

R(x) = ∫(x-μ)^2*f(x)dx

Where μ is the decision or action being taken. In this case, μ would be the value of X that we choose.

To find the minimized risk, we need to minimize the risk function R(x) with respect to μ. This can be done by taking the derivative of R(x) with respect to μ and setting it equal to 0. Solving for μ will give us the value that minimizes the risk.

I hope this helps clarify the concept of minimized risk for this problem. Let me know if you have any further questions.