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I changed minor typos only.
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Xi'an
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When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of n$n$ random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of n random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of $n$ random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

When modelling a sample $(x_1,\ldots,x_n)$ as an iid$i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of an random variablevariables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:   

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$ The

The concept of $n$ realisationrealizations of a single random variable is a shortcut asthat is not well-defined because one cannot handle independence with a single random variable.

When modelling a sample $(x_1,\ldots,x_n)$ as an iid sample from a given distribution $F$, the correct modelling is to see this sample as the realisation of a random variable $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:  $$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$ The concept of $n$ realisation of a single random variable is a shortcut as is not well-defined because one cannot handle independence with a single random variable.

When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of n random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$: 

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

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Xi'an
  • 107.7k
  • 13
  • 190
  • 676

When modelling a sample $(x_1,\ldots,x_n)$ as an iid sample from a given distribution $F$, the correct modelling is to see this sample as the realisation of a random variable $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$: $$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$ The concept of $n$ realisation of a single random variable is a shortcut as is not well-defined because one cannot handle independence with a single random variable.