Conditional Expectation problem

[JGalway]

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.

 

[Euge]

Welcome, JGalway! (Wave)

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

$$\frac{\int_1^2 x\cdot \dfrac{x}{4}\, dx + \int_2^\infty x\cdot \dfrac{4}{x^3}\, dx}{\int_1^2 \dfrac{x}{4}\, dx + \int_2^\infty \dfrac{4}{x^3}\, dx}$$

 

[JGalway]

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.

 

[MarkFL]

...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)