Question

Define the expectation of a continuous random variable.

Answer 1

Before defining expectation of continuous random variable , I should definitely what is the meaning of continuous random variable.
Continuous random variable :- A continuous random variable is a random variable where the data can take infinitely many values.

Now, Let's understand what is expectation ?
Take an example :- in case of dice , I said expectation value of dice is 3.5
How's it ? , Actually expectation is not other than mean of terms .
In dice terms are 1 , 2 , 3, 4 , 5 , 6
So, mean of it = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5
Hence, expectation = 3.5
Now, in case of continuous random variable expectation is given by
E(X) = $\int\limits^a_b{xf(x)}\,dx$ , where [a,b] is domain of f(x)
And f(x) is probability density function .