Change of variable for integral formula…why?
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I'm reading a book on mathematical statistics and it states two properties of pdfs.
1) $f_X(x)geq 0$
2) $int_-infty^inftyf_X(t)dt = 1$
My question is this. Why is the integration variable a t, not an x in the second equation?
Thanks
calculus random-variables
asked Aug 23 at 5:16
Bucephalus
446214
add a comment |Â
up vote
0
down vote
favorite
I'm reading a book on mathematical statistics and it states two properties of pdfs.
1) $f_X(x)geq 0$
2) $int_-infty^inftyf_X(t)dt = 1$
My question is this. Why is the integration variable a t, not an x in the second equation?
Thanks
calculus random-variables
asked Aug 23 at 5:16
Bucephalus
446214
What's the difference anyway?
– Vim
Aug 23 at 5:20
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm reading a book on mathematical statistics and it states two properties of pdfs.
1) $f_X(x)geq 0$
2) $int_-infty^inftyf_X(t)dt = 1$
My question is this. Why is the integration variable a t, not an x in the second equation?
Thanks
calculus random-variables
asked Aug 23 at 5:16
Bucephalus
446214
I'm reading a book on mathematical statistics and it states two properties of pdfs.
1) $f_X(x)geq 0$
2) $int_-infty^inftyf_X(t)dt = 1$
My question is this. Why is the integration variable a t, not an x in the second equation?
Thanks
calculus random-variables
asked Aug 23 at 5:16
Bucephalus
446214
asked Aug 23 at 5:16
Bucephalus
446214
asked Aug 23 at 5:16
Bucephalus
446214
asked Aug 23 at 5:16
Bucephalus
446214
446214
What's the difference anyway?
– Vim
Aug 23 at 5:20
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29
add a comment |Â
What's the difference anyway?
– Vim
Aug 23 at 5:20
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29
What's the difference anyway?
– Vim
Aug 23 at 5:20
What's the difference anyway?
– Vim
Aug 23 at 5:20
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
This is only to close the question.
Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.
Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.
If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $mathcal F_F(f)$ and there would be confusion as to what is what.
I should add that $t$ is varying from $-infty$ to $infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This is only to close the question.
Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.
Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.
If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $mathcal F_F(f)$ and there would be confusion as to what is what.
I should add that $t$ is varying from $-infty$ to $infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
add a comment |Â
up vote
2
down vote
accepted
This is only to close the question.
Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.
Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.
If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $mathcal F_F(f)$ and there would be confusion as to what is what.
I should add that $t$ is varying from $-infty$ to $infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This is only to close the question.
Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.
Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.
If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $mathcal F_F(f)$ and there would be confusion as to what is what.
I should add that $t$ is varying from $-infty$ to $infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
This is only to close the question.
Absolutely no mathematical motivation. Probably to remove confusion with the $X$. But remember , that choice of notation, especially while writing a book, must be very careful, since one requires it to be consistently maintained throughout the book.
Therefore, this notation may by repeated through the book : as you read on, you will realize the role of t instead of x.
If notation is cluttered or variables are not chosen nicely, you could land in trouble. For example, I could write $f_X(t)$ as $mathcal F_F(f)$ and there would be confusion as to what is what.
I should add that $t$ is varying from $-infty$ to $infty$. My guess is that $t$ usually varies over the reals in the book, as an integration variable. I have seen this in other books on probability as well.
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
answered Aug 23 at 5:30
ðÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
33.1k22665
33.1k22665
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
add a comment |Â
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
This was an important answer to me as we all, at varying points, want to understand why something is done, not just accept that something is done. Thanks @Actoh. yep, in fact the author goes on to state, that if the pdf has these two properties then it is for a continuous random variabl.
– Bucephalus
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
In other words: nobody cares; the variables are dummies.
– Sean Roberson
Aug 23 at 5:37
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
You are welcome! Having read some probability, I have also seen the same notation being stuck to by professors in class.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:38
add a comment |Â
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What's the difference anyway?
– Vim
Aug 23 at 5:20
I think this prevents confusion with the $X$.
– Ã°ÑÂтþý òіûûð þûþф üÑÂûûñÑÂрó
Aug 23 at 5:23
hhhmmm, so there doesn't seem to be some mathematical motivation behind it, just more readability motivation.
– Bucephalus
Aug 23 at 5:26
Ok, well put that as an answer then @Actoh, and I will accept it if that's ok with you.
– Bucephalus
Aug 23 at 5:29