High School Expectation of probability density function

[songoku]

E(X) of probability density function f(x) is \int x f(x) dx

E(X²) of probability density function f(x) is \int x^2 f(x) dx

Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ?

I tried to find E(5 + 10X) from pdf. I did two ways:
1. I found E(X) then using linear combination of random variable, E (5 + 10X) = 5 + 10 E(X)

2. Using integration, \int (5 + 10x) f(x) dx

I got two different answers. Which one is correct and why?

Thanks

 

[PeroK]

E(X) of probability density function f(x) is \int x f(x) dx

E(X²) of probability density function f(x) is \int x^2 f(x) dx

Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ?

I tried to find E(5 + 10X) from pdf. I did two ways:
1. I found E(X) then using linear combination of random variable, E (5 + 10X) = 5 + 10 E(X)

2. Using integration, \int (5 + 10x) f(x) dx

I got two different answers. Which one is correct and why?

Thanks

They are both correct. You must have made a mistake integrating.

 

[PeroK]

E(X) of probability density function f(x) is \int x f(x) dx

E(X²) of probability density function f(x) is \int x^2 f(x) dx

Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ?

Yes. In fact, you could take that to be the definition of the expected value of some function $g(x)$. Then, $E(x)$ and $E(x^2)$ are specific cases of $g(x)$.

 

[songoku]

I re-checked so many times and apparently the mistake was I wrote the question wrongly

Thank you so much perok