# Probability - Expectation calculation for a function.

A company manufactures metal poles. Suppose the length of a pole is a random variable X, with mean $\mu _X$ and probability density function $f_X(x)$ . Poles are cut to obtain an exact length 𝐿. If the initial length of the pole is less than 𝐿, the entire pole is lost. If it is greater than 𝐿, the pole will be cut down to 𝐿, and the section left over is lost. We are interested in the random variable 𝑌, defined as the length of each piece lost.

Sketch the graph of the function 𝑔 that maps the pole length 𝑥 to the lost length 𝑦, and so derive $\mu _Y$ = 𝔼(𝑌) as a function of $f_X(x)$ and $\mu _X$.

Suppose that 𝑋 follows a normal distribution with mean $\mu _X$ and variance $\sigma _X ^2$ . Show that there exists a value $\mu ^*$ of $\mu _X$ that minimizes $\mu _Y$.

## Answer

**Answers can be viewed only if**

- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.

- answered
- 334 views
- $5.00

### Related Questions

- Probabilty question
- Statistics and Probability
- Figuring out the maths for the probability of two adopted teens randomly being matched as pen pals in 2003
- Probability/Analysis Question
- What is the probability that the last person to board an airplane gets to sit in their assigned seat?
- Calculating P values from data.
- Summation of Catalan Convolution
- Find Mean, Variance, and Distribution of Recursively Defined Sequence of Random Variables