Finding the variance of speeds

Question

This is a question from my Statistics textbook which I am currently stuck on. I have approached the question in a couple ways but each time I have been incorrect.

A summary of the speeds, x kilometers per hour, of 22 cars passing a certain point gave the following information:

E(x-50) = 81.4 and E(x-50)^2 = 671.0

How would I find the variance of the speeds and thus find the value of Ex^2? What are the steps when finding this? Any help is appreciated.

Answer 1

Expectation is linear, so $\Bbb E(x-50)=\Bbb E(x)-\Bbb E(50)=\Bbb E(x)-50$, and from this you can easily get $\Bbb E(x)$. Now multiply out $(x-50)^2$ and apply linearity of expectation to the resulting expression, substituting the value of $\Bbb E(x)$ from the first step, and solve for $\Bbb E(x^2)$.