What is the mathematical interpretation of random variable equation?












1












$begingroup$


Following is the equation:
$$X = Y + Z.$$
$X, Y, Z$ are random variables. This is a random variable equation. What is the meaning of this equation? Does it mean that if you take any value of $Y$ and any value of $Z$, it should equal $X$? Or does it mean that when $Y$ and $Z$ are added, then $Y+Z$ will have same distribution as $X$?






probability probability-theory probability-distributions random-variables







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  • $begingroup$
    It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
    $endgroup$
    – gd1035
    Dec 7 '18 at 2:25










  • $begingroup$
    The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
    $endgroup$
    – herb steinberg
    Dec 7 '18 at 2:26










  • $begingroup$
    It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
    $endgroup$
    – Will M.
    Dec 7 '18 at 2:38






  • 3




    $begingroup$
    It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
    $endgroup$
    – littleO
    Dec 7 '18 at 2:48






  • 1




    $begingroup$
    @herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
    $endgroup$
    – Did
    Dec 7 '18 at 9:38
















1












$begingroup$


Following is the equation:
$$X = Y + Z.$$
$X, Y, Z$ are random variables. This is a random variable equation. What is the meaning of this equation? Does it mean that if you take any value of $Y$ and any value of $Z$, it should equal $X$? Or does it mean that when $Y$ and $Z$ are added, then $Y+Z$ will have same distribution as $X$?






probability probability-theory probability-distributions random-variables







share|cite|improve this question











$endgroup$












  • $begingroup$
    It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
    $endgroup$
    – gd1035
    Dec 7 '18 at 2:25










  • $begingroup$
    The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
    $endgroup$
    – herb steinberg
    Dec 7 '18 at 2:26










  • $begingroup$
    It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
    $endgroup$
    – Will M.
    Dec 7 '18 at 2:38






  • 3




    $begingroup$
    It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
    $endgroup$
    – littleO
    Dec 7 '18 at 2:48






  • 1




    $begingroup$
    @herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
    $endgroup$
    – Did
    Dec 7 '18 at 9:38














1












1








1


1



$begingroup$


Following is the equation:
$$X = Y + Z.$$
$X, Y, Z$ are random variables. This is a random variable equation. What is the meaning of this equation? Does it mean that if you take any value of $Y$ and any value of $Z$, it should equal $X$? Or does it mean that when $Y$ and $Z$ are added, then $Y+Z$ will have same distribution as $X$?






probability probability-theory probability-distributions random-variables







share|cite|improve this question











$endgroup$




Following is the equation:
$$X = Y + Z.$$
$X, Y, Z$ are random variables. This is a random variable equation. What is the meaning of this equation? Does it mean that if you take any value of $Y$ and any value of $Z$, it should equal $X$? Or does it mean that when $Y$ and $Z$ are added, then $Y+Z$ will have same distribution as $X$?






probability probability-theory probability-distributions random-variables




probability probability-theory probability-distributions random-variables






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 2:36









timtfj

1,746418




1,746418










asked Dec 7 '18 at 2:09









ShiranShiran

82




82












  • $begingroup$
    It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
    $endgroup$
    – gd1035
    Dec 7 '18 at 2:25










  • $begingroup$
    The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
    $endgroup$
    – herb steinberg
    Dec 7 '18 at 2:26










  • $begingroup$
    It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
    $endgroup$
    – Will M.
    Dec 7 '18 at 2:38






  • 3




    $begingroup$
    It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
    $endgroup$
    – littleO
    Dec 7 '18 at 2:48






  • 1




    $begingroup$
    @herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
    $endgroup$
    – Did
    Dec 7 '18 at 9:38


















  • $begingroup$
    It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
    $endgroup$
    – gd1035
    Dec 7 '18 at 2:25










  • $begingroup$
    The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
    $endgroup$
    – herb steinberg
    Dec 7 '18 at 2:26










  • $begingroup$
    It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
    $endgroup$
    – Will M.
    Dec 7 '18 at 2:38






  • 3




    $begingroup$
    It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
    $endgroup$
    – littleO
    Dec 7 '18 at 2:48






  • 1




    $begingroup$
    @herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
    $endgroup$
    – Did
    Dec 7 '18 at 9:38
















$begingroup$
It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
$endgroup$
– gd1035
Dec 7 '18 at 2:25




$begingroup$
It is pretty common to add random variables together to get a new random variable. For example, the binomial distribution is a sum of independent Bernoulli trials. So I think your second interpretation is correct.
$endgroup$
– gd1035
Dec 7 '18 at 2:25












$begingroup$
The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
$endgroup$
– herb steinberg
Dec 7 '18 at 2:26




$begingroup$
The equation $X=Y+Z$ usually means $X$ is a random variable defined as the sum of $Y$ and $Z$. In other contexts it may mean that a particular value of $X$ is the sum of a particular value of $Y+Z$.and you might ask what is the probability of such an equality. As you can see it all depends on context.
$endgroup$
– herb steinberg
Dec 7 '18 at 2:26












$begingroup$
It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
$endgroup$
– Will M.
Dec 7 '18 at 2:38




$begingroup$
It means that the random result $X$ equals the sum of the random results $Y$ and $Z.$
$endgroup$
– Will M.
Dec 7 '18 at 2:38




3




3




$begingroup$
It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
$endgroup$
– littleO
Dec 7 '18 at 2:48




$begingroup$
It does not simply mean that $X$ and $Y + Z$ have the same distribution. It means that $X$ is $Y + Z$.
$endgroup$
– littleO
Dec 7 '18 at 2:48




1




1




$begingroup$
@herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
$endgroup$
– Did
Dec 7 '18 at 9:38




$begingroup$
@herbsteinberg "As you can see it all depends on context" No it does not. Why mislead the OP?
$endgroup$
– Did
Dec 7 '18 at 9:38










1 Answer
1






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$begingroup$

Random variables are defined as measurable functions from a probability space $Omega$ into some target space. Without any further context, I interpret that equality as an equality of functions, e.g. $X(omega) = Y(omega) + Z(omega)$ for every $omega in Omega$. Quite often in probability, we only care about equality on a set of measure 1, and it is common to see $X = Y+Z$ a.s. where a.s. is an abbreviation for almost surely, which means that the functions $X$ and $Y+Z$ might differ on a tiny set of measure 0, but we don't care.






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  • $begingroup$
    Thanks for the explanation...
    $endgroup$
    – Shiran
    Dec 8 '18 at 2:49











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1 Answer
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1 Answer
1






active

oldest

votes









active

oldest

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active

oldest

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1












$begingroup$

Random variables are defined as measurable functions from a probability space $Omega$ into some target space. Without any further context, I interpret that equality as an equality of functions, e.g. $X(omega) = Y(omega) + Z(omega)$ for every $omega in Omega$. Quite often in probability, we only care about equality on a set of measure 1, and it is common to see $X = Y+Z$ a.s. where a.s. is an abbreviation for almost surely, which means that the functions $X$ and $Y+Z$ might differ on a tiny set of measure 0, but we don't care.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for the explanation...
    $endgroup$
    – Shiran
    Dec 8 '18 at 2:49
















1












$begingroup$

Random variables are defined as measurable functions from a probability space $Omega$ into some target space. Without any further context, I interpret that equality as an equality of functions, e.g. $X(omega) = Y(omega) + Z(omega)$ for every $omega in Omega$. Quite often in probability, we only care about equality on a set of measure 1, and it is common to see $X = Y+Z$ a.s. where a.s. is an abbreviation for almost surely, which means that the functions $X$ and $Y+Z$ might differ on a tiny set of measure 0, but we don't care.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks for the explanation...
    $endgroup$
    – Shiran
    Dec 8 '18 at 2:49














1












1








1





$begingroup$

Random variables are defined as measurable functions from a probability space $Omega$ into some target space. Without any further context, I interpret that equality as an equality of functions, e.g. $X(omega) = Y(omega) + Z(omega)$ for every $omega in Omega$. Quite often in probability, we only care about equality on a set of measure 1, and it is common to see $X = Y+Z$ a.s. where a.s. is an abbreviation for almost surely, which means that the functions $X$ and $Y+Z$ might differ on a tiny set of measure 0, but we don't care.






share|cite|improve this answer









$endgroup$



Random variables are defined as measurable functions from a probability space $Omega$ into some target space. Without any further context, I interpret that equality as an equality of functions, e.g. $X(omega) = Y(omega) + Z(omega)$ for every $omega in Omega$. Quite often in probability, we only care about equality on a set of measure 1, and it is common to see $X = Y+Z$ a.s. where a.s. is an abbreviation for almost surely, which means that the functions $X$ and $Y+Z$ might differ on a tiny set of measure 0, but we don't care.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 7 '18 at 5:47









zoidbergzoidberg

1,065113




1,065113












  • $begingroup$
    Thanks for the explanation...
    $endgroup$
    – Shiran
    Dec 8 '18 at 2:49


















  • $begingroup$
    Thanks for the explanation...
    $endgroup$
    – Shiran
    Dec 8 '18 at 2:49
















$begingroup$
Thanks for the explanation...
$endgroup$
– Shiran
Dec 8 '18 at 2:49




$begingroup$
Thanks for the explanation...
$endgroup$
– Shiran
Dec 8 '18 at 2:49


















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