I'm reading a book on statistics and the definition given of a random sample doesn't make much sense to me.
Here's the idea that I have of a random sample: in a given population, any sample of size $n$ has an equal probability of being chosen. Let $x'_1$,..., $x'_n$ be the values of the first,..., and last element of the first sample of size $n$; let $x''_1$,..., $x''_n$ be the values of the first,..., and last element of the second sample of size $n$, and so on and so forth. Let $X_1$ be the random variable associated with values $x'_1$, $x''_1$, etc.; $X_2$ be the random variable associated with the values $x'_2$, $x''_2$, etc.
If $X_1$,..., $X_m$ are independent variables and each $X_i$ has the same probability distribution then ($X_1$,..., $X_m$) are a random sample.
Am I wrong or right?