I'm trying to understand the part of this book, page 276 which explains about sample mean:
In statistical inference, a central problem is how to use data to estimate unknown parameters of a distribution, or functions of unknown parameters. It is especially common to want to estimate the mean and variance of a distribution. If the data are i.i.d. random variables $X_1,... ,X_n$ where the mean $E(X_j)$ is unknown, then the most obvious way to estimate the mean is simply to average the $X_j$ , taking the arithmetic mean.
For example, if the observed data are $3, 1, 1, 5$, then a simple, natural way to estimate the mean of the distribution that generated the data is to use $(3+1+1+5)/4 = 2.5$. This is called the sample mean.
My question is Instead of saying "... i.i.d. random variables $X_1,... ,X_n$ where the mean $E(X_j)$ is unknown" can I write "let X be a random variable and take the $n$ outputs of $X$, where E(X) is unknown". In another words, can I see these $X_1,... ,X_n$ as only an one random variable? since the $X_1,... ,X_n$ are independent and identically distributed?