How to interpret this statement? [closed]

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A random variable $X$ is said to be continuous if $P(X = x) = 0$ for all single points $x ∈ X$.

Source： https://arxiv.org/pdf/1701.04862.pdf#page=16

1) Why should $P(X=x)$ be 0 for all single points to make the random variable $X$ continuous? What does it mean for $P(X=x)$ to be 0?

2) Can it be true that:

A random variable $X$ is said to be continuous if and only if $P(X = x) = 0$ for all single points $x ∈ X$.

It seems obvious that if $P(X = x) = 0$ for all single points $x ∈ X$ a random varible $X$ would be said to be continuous.

3) And if for some points the probabilities are not 0, could that variable be possibly continuous too?

probability-theory

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edited Jul 19 at 9:04

asked Jul 18 at 12:01

lerner

242113

closed as off-topic by Did, Isaac Browne, Xander Henderson, Jose Arnaldo Bebita Dris, Ethan Bolker Jul 19 at 22:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Isaac Browne, Jose Arnaldo Bebita Dris

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  What about this statement is unclear to you?
  – Mees de Vries
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
  – Henning Makholm
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
  – David C. Ullrich
  Jul 18 at 13:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How is the update addressing @Mees' query?
  – Did
  Jul 19 at 2:04
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
  – littleO
  Jul 19 at 2:19
  
  

  
  
  
  

 | 
show 4 more comments

up vote
0
down vote

favorite

A random variable $X$ is said to be continuous if $P(X = x) = 0$ for all single points $x ∈ X$.

Source： https://arxiv.org/pdf/1701.04862.pdf#page=16

1) Why should $P(X=x)$ be 0 for all single points to make the random variable $X$ continuous? What does it mean for $P(X=x)$ to be 0?

2) Can it be true that:

A random variable $X$ is said to be continuous if and only if $P(X = x) = 0$ for all single points $x ∈ X$.

It seems obvious that if $P(X = x) = 0$ for all single points $x ∈ X$ a random varible $X$ would be said to be continuous.

3) And if for some points the probabilities are not 0, could that variable be possibly continuous too?

probability-theory

share|cite|improve this question

edited Jul 19 at 9:04

asked Jul 18 at 12:01

lerner

242113

closed as off-topic by Did, Isaac Browne, Xander Henderson, Jose Arnaldo Bebita Dris, Ethan Bolker Jul 19 at 22:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Isaac Browne, Jose Arnaldo Bebita Dris

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  What about this statement is unclear to you?
  – Mees de Vries
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
  – Henning Makholm
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
  – David C. Ullrich
  Jul 18 at 13:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How is the update addressing @Mees' query?
  – Did
  Jul 19 at 2:04
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
  – littleO
  Jul 19 at 2:19
  
  

  
  
  
  

 | 
show 4 more comments

up vote
0
down vote

favorite

up vote
0
down vote

favorite

A random variable $X$ is said to be continuous if $P(X = x) = 0$ for all single points $x ∈ X$.

Source： https://arxiv.org/pdf/1701.04862.pdf#page=16

1) Why should $P(X=x)$ be 0 for all single points to make the random variable $X$ continuous? What does it mean for $P(X=x)$ to be 0?

2) Can it be true that:

A random variable $X$ is said to be continuous if and only if $P(X = x) = 0$ for all single points $x ∈ X$.

It seems obvious that if $P(X = x) = 0$ for all single points $x ∈ X$ a random varible $X$ would be said to be continuous.

3) And if for some points the probabilities are not 0, could that variable be possibly continuous too?

probability-theory

share|cite|improve this question

edited Jul 19 at 9:04

asked Jul 18 at 12:01

lerner

242113

A random variable $X$ is said to be continuous if $P(X = x) = 0$ for all single points $x ∈ X$.

Source： https://arxiv.org/pdf/1701.04862.pdf#page=16

1) Why should $P(X=x)$ be 0 for all single points to make the random variable $X$ continuous? What does it mean for $P(X=x)$ to be 0?

2) Can it be true that:

A random variable $X$ is said to be continuous if and only if $P(X = x) = 0$ for all single points $x ∈ X$.

It seems obvious that if $P(X = x) = 0$ for all single points $x ∈ X$ a random varible $X$ would be said to be continuous.

3) And if for some points the probabilities are not 0, could that variable be possibly continuous too?

probability-theory

share|cite|improve this question

edited Jul 19 at 9:04

asked Jul 18 at 12:01

lerner

242113

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 9:04

edited Jul 19 at 9:04

edited Jul 19 at 9:04

asked Jul 18 at 12:01

lerner

242113

asked Jul 18 at 12:01

lerner

242113

asked Jul 18 at 12:01

lerner

242113

242113

closed as off-topic by Did, Isaac Browne, Xander Henderson, Jose Arnaldo Bebita Dris, Ethan Bolker Jul 19 at 22:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Isaac Browne, Jose Arnaldo Bebita Dris

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by Did, Isaac Browne, Xander Henderson, Jose Arnaldo Bebita Dris, Ethan Bolker Jul 19 at 22:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Isaac Browne, Jose Arnaldo Bebita Dris

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  What about this statement is unclear to you?
  – Mees de Vries
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
  – Henning Makholm
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
  – David C. Ullrich
  Jul 18 at 13:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How is the update addressing @Mees' query?
  – Did
  Jul 19 at 2:04
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
  – littleO
  Jul 19 at 2:19
  
  

  
  
  
  

 | 
show 4 more comments

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  What about this statement is unclear to you?
  – Mees de Vries
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
  – Henning Makholm
  Jul 18 at 12:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
  – David C. Ullrich
  Jul 18 at 13:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How is the update addressing @Mees' query?
  – Did
  Jul 19 at 2:04
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
  – littleO
  Jul 19 at 2:19
  
  

  
  
  
  

3

3

What about this statement is unclear to you?
– Mees de Vries
Jul 18 at 12:07

What about this statement is unclear to you?
– Mees de Vries
Jul 18 at 12:07

4

4

It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
– Henning Makholm
Jul 18 at 12:07

It means exactly what it says, and is as far as I can tell consistent with the common use of the term "continuous random variable". Do you have more than one possible interpretation of the definition in mind?
– Henning Makholm
Jul 18 at 12:07

1

1

Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
– David C. Ullrich
Jul 18 at 13:55

Does it really say "for all single points $xin X$"? That makes no sense - should be "for all $xinBbb R$". Given that, it's hard to see how there's anything to "interpret"...
– David C. Ullrich
Jul 18 at 13:55

How is the update addressing @Mees' query?
– Did
Jul 19 at 2:04

How is the update addressing @Mees' query?
– Did
Jul 19 at 2:04

2

2

Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
– littleO
Jul 19 at 2:19

Roughly speaking, a continuous random variable is a random variable that can take on a continuous range of values, such as any value in an interval $[a,b]$ for example. For example, throw a dart at a dart board, and let $X$ be the distance from where the dart landed to the bullseye. You can see it's utterly unlikely that $X$ would be exactly equal to, say, $1.7$. So $P(X = 1.7) = 0$. The same goes for any other specific number. In contrast, for a discrete random variable there are only a finite (or countable) number of possible values, each with positive probability.
– littleO
Jul 19 at 2:19

 | 
show 4 more comments

1 Answer
1

active

oldest

votes

up vote
1
down vote



accepted

The idea is that broadly, we can break random variables into discrete and continuous random variables.

Discrete random variables are ones that can only take on discrete values. Like rolling a die. You can get the values $1,2,3,4,5,6$. Or the number of people at a party. You can only have positive integer number of people. You can have $1,2,3,...$ people at the party. Or the amount of money you can win on a slot machine. The least amount you can win is say, $.01$.

Continuous random variables are ones where you can have any value in an interval. Say, someone's height. You can be exactly $5.0001294024902390...$ feet tall. Or the age when you die. You can be $80.2323999...$ years old. The idea is that any individual outcome is "extremely unlikely". So rather than asking about what is the probability of being exactly $5.00...$ feet tall we can ask "what's the probability that you're between $5$ and $5.1$ feet tall?". In these cases, we say the probability of being exactly $5$ feet tall is $0$.

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

The idea is that broadly, we can break random variables into discrete and continuous random variables.

Discrete random variables are ones that can only take on discrete values. Like rolling a die. You can get the values $1,2,3,4,5,6$. Or the number of people at a party. You can only have positive integer number of people. You can have $1,2,3,...$ people at the party. Or the amount of money you can win on a slot machine. The least amount you can win is say, $.01$.

Continuous random variables are ones where you can have any value in an interval. Say, someone's height. You can be exactly $5.0001294024902390...$ feet tall. Or the age when you die. You can be $80.2323999...$ years old. The idea is that any individual outcome is "extremely unlikely". So rather than asking about what is the probability of being exactly $5.00...$ feet tall we can ask "what's the probability that you're between $5$ and $5.1$ feet tall?". In these cases, we say the probability of being exactly $5$ feet tall is $0$.

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

add a comment | 

up vote
1
down vote



accepted

The idea is that broadly, we can break random variables into discrete and continuous random variables.

Discrete random variables are ones that can only take on discrete values. Like rolling a die. You can get the values $1,2,3,4,5,6$. Or the number of people at a party. You can only have positive integer number of people. You can have $1,2,3,...$ people at the party. Or the amount of money you can win on a slot machine. The least amount you can win is say, $.01$.

Continuous random variables are ones where you can have any value in an interval. Say, someone's height. You can be exactly $5.0001294024902390...$ feet tall. Or the age when you die. You can be $80.2323999...$ years old. The idea is that any individual outcome is "extremely unlikely". So rather than asking about what is the probability of being exactly $5.00...$ feet tall we can ask "what's the probability that you're between $5$ and $5.1$ feet tall?". In these cases, we say the probability of being exactly $5$ feet tall is $0$.

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

The idea is that broadly, we can break random variables into discrete and continuous random variables.

Discrete random variables are ones that can only take on discrete values. Like rolling a die. You can get the values $1,2,3,4,5,6$. Or the number of people at a party. You can only have positive integer number of people. You can have $1,2,3,...$ people at the party. Or the amount of money you can win on a slot machine. The least amount you can win is say, $.01$.

Continuous random variables are ones where you can have any value in an interval. Say, someone's height. You can be exactly $5.0001294024902390...$ feet tall. Or the age when you die. You can be $80.2323999...$ years old. The idea is that any individual outcome is "extremely unlikely". So rather than asking about what is the probability of being exactly $5.00...$ feet tall we can ask "what's the probability that you're between $5$ and $5.1$ feet tall?". In these cases, we say the probability of being exactly $5$ feet tall is $0$.

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

The idea is that broadly, we can break random variables into discrete and continuous random variables.

Discrete random variables are ones that can only take on discrete values. Like rolling a die. You can get the values $1,2,3,4,5,6$. Or the number of people at a party. You can only have positive integer number of people. You can have $1,2,3,...$ people at the party. Or the amount of money you can win on a slot machine. The least amount you can win is say, $.01$.

Continuous random variables are ones where you can have any value in an interval. Say, someone's height. You can be exactly $5.0001294024902390...$ feet tall. Or the age when you die. You can be $80.2323999...$ years old. The idea is that any individual outcome is "extremely unlikely". So rather than asking about what is the probability of being exactly $5.00...$ feet tall we can ask "what's the probability that you're between $5$ and $5.1$ feet tall?". In these cases, we say the probability of being exactly $5$ feet tall is $0$.

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

share|cite|improve this answer

share|cite|improve this answer

answered Jul 19 at 2:24

user223391

answered Jul 19 at 2:24

user223391

answered Jul 19 at 2:24

user223391

add a comment | 

add a comment |