MHB Conditional Expectation problem

AI Thread Summary
The discussion revolves around calculating the expected length of a telephone conversation given a specific probability density function. The initial attempt at the calculation yielded an incorrect result of 5.55555 minutes, while the correct answer is stated to be 2.95 minutes. A clarification was provided regarding the correct expression for conditional expectation, emphasizing the proper integration setup. Additionally, participants discussed the use of LaTeX for mathematical formatting, confirming its compatibility with other math sites and the availability of tools for easier usage on the forum. The conversation highlights both the mathematical challenge and the supportive learning environment of the forum.
JGalway
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Q The amount of time (in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density:


$$f(x) = \begin{cases} \dfrac{x}{4}&\text{for $0 < x \le 2$}\\
\dfrac{4}{x^3}&\text{for $x > 2$}\\
0&\text{elsewhere}\end{cases}$$


with reference to part (b) of Exercise $4.59$, find the expected length of one of these telephone conversations that has lasted for 1 minute.

A: The formula from $4.59$(b) is $$E[u(x)|a<x \le b]= \frac{\int_a^b u(x)f(x)\, dx}{\int_a^b f(x)\, dx}$$

I tried $$E[x|x \ge 1]= \frac{\int_1^2 x(x/4)\, dx}{\int_1^2 x/4\, dx} + \frac{\int_2^\infty x(4/x^3)\, dx}{\int_2^\infty 4/x^3 \, dx}
= \frac{14/6}{9/6}+4=\text{5.55555 minutes}$$

but the back of the books says the answer is $2.95$ mins so i don't know where i went wrong.
 
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Welcome, JGalway! (Wave)

The expression you wrote for $E[X|X \ge 1]$ should instead be

$$\frac{\int_1^2 x\cdot \dfrac{x}{4}\, dx + \int_2^\infty x\cdot \dfrac{4}{x^3}\, dx}{\int_1^2 \dfrac{x}{4}\, dx + \int_2^\infty \dfrac{4}{x^3}\, dx}$$
 
Euge said:
Welcome, JGalway! (Wave)

The expression you wrote for $E[X|X \ge 1]$ should instead be

$$\frac{\int_1^2 x\cdot \dfrac{x}{4}\, dx + \int_2^\infty x\cdot \dfrac{4}{x^3}\, dx}{\int_1^2 \dfrac{x}{4}\, dx + \int_2^\infty \dfrac{4}{x^3}\, dx}$$
Thanks for that, I sometimes make silly mistakes like that when I get tired.
Also is the maths formatting used here the same as most sites(showing integral sign,etc)? Just not sure if I want to learn it just for this site.
 
JGalway said:
...Also is the maths formatting used here the same as most sites(showing integral sign,etc)? Just not sure if I want to learn it just for this site.

Yes, we use $\LaTeX$ powered by MathJax, which is what you'll find on most other math sites. The only difference is, unlike other sites, we provide you with easy to use tools for creating the code/markup for displaying math expressions, and a means of previewing it in real time before putting it in your post. Thus, MHB is the perfect environment to learn how to use $\LaTeX$, which you will find useful pretty much everywhere else. (Yes)
 
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