Difference between random variable and observation

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A random variable X is defined by its probability density function (p.d.f.) fX(x;θ), which depends on a deterministic parameter θ. Each sampled observation xi from N observations can be treated as a separate random variable with the same p.d.f. fX(x;θ). This approach is valid for calculating the expected value of an estimator based on these observations. The exercise involves computing the expected value of the estimator \hat{θ}(N) = s(x1,...,xN). Therefore, treating each observation as a random variable is appropriate for the calculations.
mnb96
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Hello,
I am given a random variable X with a p.d.f. fX(x;\theta) (depending on a certain deterministic parameter \theta) and I want to consider N sampled observations of that variable: x1,...,xN.
Is it correct to consider each observation as a separate random variable xi with the same pdf fX(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,...,xN are the sampled observations from the distribution of X.
 
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For your calculation, yes.
 

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