Difference between random variable and observation

[mnb96]

Hello,
I am given a random variable X with a p.d.f. f_X(x;$\theta$) (depending on a certain deterministic parameter $\theta$) and I want to consider N sampled observations of that variable: x₁,...,x_N.
Is it correct to consider each observation as a separate random variable x_i with the same pdf f_X(x,$\theta$) associated with it?

I am asking this question because I have got an exercise in which I have to compute the expected value of a given estimator:

$$\hat{\theta}(N)=s(x_1,\ldots,x_N)$$

where x₁,...,x_N are the sampled observations from the distribution of X.

 

[mathman]

For your calculation, yes.