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Elvis
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So, let’s address this question first: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meantmeans by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law than twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution hasas $2 X_1$).

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law than twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

So, let’s address this question first: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy means by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law than twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution as $2 X_1$).

edited body
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Elvis
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So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law thatthan twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law that twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law than twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

added 496 characters in body
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Elvis
  • 12.9k
  • 43
  • 59

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law that twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

So, let’s address this question: ''What is the basic difference between an algebraic variable and a Random Variable?''

A random variable $X$ is not at all an algebraic variable. Formally, it is defined as a function from a probability space $\Omega$ to $\mathbb R$.

OK... What that really means is that you perform random experiments (eg throwing a dice, choosing a random human), and you make measures on these experiments (eg number on dice upper face, height, sex, cholesterol level of the human). The set $\Omega$ is the set of all possible experiments. On a particular experiment $\omega\in\Omega$, you make a measure $X(\omega)$: that’s why formally it is a function from $\Omega$ to $\mathbb R$.

Now in general we totally forget about $\Omega$. The random variables are defined in term of their probability law. In the case of a fair dice, you just say

  • $\mathbb P(X = k) = {1\over 6}$ for $k=1,\dots,6$ (the probability of $X$ equal to $k$ is 1/6 for $k$ from 1 to 6),

instead of

  • $\mathbb P\left( \bigl\{ \omega \in \Omega \ : \ X(\omega) = k\bigr\}\right)$ (the set of dice throws on which the measure $X$ — upper face — is $k$ has probability 1/6)...

It’s simpler. You can even totally avoid bothering the students with $\Omega$.

I hope this sheds some kind of light.

Now what this guy meant by $X + X \ne 2X$ isn’t that the sum of such a measure with itself is not twice this measure — unfortunately, it is what he writes. What he means is that the sum of two such measures, performed on different experiments, has not the same law that twice a measure. This could be written as $X_1 \sim X_2 \not\Rightarrow X_1 + X_2 \sim 2X_1$ (the fact that $X_1$ and $X_2$ have the same distribution does not imply that $X_1 + X_2$ has the same distribution has $2 X_1$).

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Elvis
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