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Assume I choose $n$ random integers such that the last digit is uniformly distributed. What is the distribution of the last digit of the sum?
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Say that I sample $n$ random integers from some random variable $X$. The distribution has the last digit of the integer uniformly distributed. I then take the samples and add them
$$
Y = x_1+x_2+x_3 + ... + x_n
$$
What is the distribution of the last digit of $Y$? I want to also say uniform, but I'm not sure
probability combinatorics statistics probability-distributions uniform-distribution
asked Jul 17 at 16:03
wjmccann
571117
add a comment |Â
up vote
1
down vote
favorite
Say that I sample $n$ random integers from some random variable $X$. The distribution has the last digit of the integer uniformly distributed. I then take the samples and add them
$$
Y = x_1+x_2+x_3 + ... + x_n
$$
What is the distribution of the last digit of $Y$? I want to also say uniform, but I'm not sure
probability combinatorics statistics probability-distributions uniform-distribution
asked Jul 17 at 16:03
wjmccann
571117
You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Say that I sample $n$ random integers from some random variable $X$. The distribution has the last digit of the integer uniformly distributed. I then take the samples and add them
$$
Y = x_1+x_2+x_3 + ... + x_n
$$
What is the distribution of the last digit of $Y$? I want to also say uniform, but I'm not sure
probability combinatorics statistics probability-distributions uniform-distribution
asked Jul 17 at 16:03
wjmccann
571117
Say that I sample $n$ random integers from some random variable $X$. The distribution has the last digit of the integer uniformly distributed. I then take the samples and add them
$$
Y = x_1+x_2+x_3 + ... + x_n
$$
What is the distribution of the last digit of $Y$? I want to also say uniform, but I'm not sure
probability combinatorics statistics probability-distributions uniform-distribution
asked Jul 17 at 16:03
wjmccann
571117
asked Jul 17 at 16:03
wjmccann
571117
asked Jul 17 at 16:03
wjmccann
571117
asked Jul 17 at 16:03
wjmccann
571117
571117
You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07
add a comment |Â
You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07
You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07
You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
accepted
Yes, it's uniform. In fact, no matter what the value of $x_1+cdots+x_n-1$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.
answered Jul 17 at 16:11
Especially Lime
19.1k22252
add a comment |Â
up vote
2
down vote
Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $mod10$. In order to prove your statement, just prove
If $X$ and $Y$ are uniform $mod10$ and independent, then $X+Y$ is uniform $mod10$ as well.
Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X mod10$ first.
answered Jul 17 at 16:08
Marcus M
8,1731847
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Yes, it's uniform. In fact, no matter what the value of $x_1+cdots+x_n-1$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.
answered Jul 17 at 16:11
Especially Lime
19.1k22252
add a comment |Â
up vote
4
down vote
accepted
Yes, it's uniform. In fact, no matter what the value of $x_1+cdots+x_n-1$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.
answered Jul 17 at 16:11
Especially Lime
19.1k22252
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Yes, it's uniform. In fact, no matter what the value of $x_1+cdots+x_n-1$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.
answered Jul 17 at 16:11
Especially Lime
19.1k22252
Yes, it's uniform. In fact, no matter what the value of $x_1+cdots+x_n-1$ is, the ten possibilities for the last digit of $x_n$ all give different values for the last digit of $Y$. Since the last digit of $x_n$ is equally likely to be any of the ten possibilities, the same is true for $Y$. You don't need to know that all your $x_i$ are uniform in the last digit; as long as at least one of them is, the answer is uniform.
answered Jul 17 at 16:11
Especially Lime
19.1k22252
answered Jul 17 at 16:11
Especially Lime
19.1k22252
answered Jul 17 at 16:11
Especially Lime
19.1k22252
answered Jul 17 at 16:11
Especially Lime
19.1k22252
19.1k22252
add a comment |Â
add a comment |Â
up vote
2
down vote
Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $mod10$. In order to prove your statement, just prove
If $X$ and $Y$ are uniform $mod10$ and independent, then $X+Y$ is uniform $mod10$ as well.
Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X mod10$ first.
answered Jul 17 at 16:08
Marcus M
8,1731847
add a comment |Â
up vote
2
down vote
Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $mod10$. In order to prove your statement, just prove
If $X$ and $Y$ are uniform $mod10$ and independent, then $X+Y$ is uniform $mod10$ as well.
Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X mod10$ first.
answered Jul 17 at 16:08
Marcus M
8,1731847
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $mod10$. In order to prove your statement, just prove
If $X$ and $Y$ are uniform $mod10$ and independent, then $X+Y$ is uniform $mod10$ as well.
Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X mod10$ first.
answered Jul 17 at 16:08
Marcus M
8,1731847
Yes, it is uniform: one good way to see this is to realize that the last digit is just the number $mod10$. In order to prove your statement, just prove
If $X$ and $Y$ are uniform $mod10$ and independent, then $X+Y$ is uniform $mod10$ as well.
Then use induction to get your statement. If you need a hint on this step, consider conditioning on $X mod10$ first.
answered Jul 17 at 16:08
Marcus M
8,1731847
answered Jul 17 at 16:08
Marcus M
8,1731847
answered Jul 17 at 16:08
Marcus M
8,1731847
answered Jul 17 at 16:08
Marcus M
8,1731847
8,1731847
add a comment |Â
add a comment |Â
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You can rephrase the question as "$x_i$ being uniformly distributed residues modulo $10$, is $x_1+ldots +x_n$ also uniformly distributed?". It doesn't matter that the $x_i$s are integers.
– Arnaud Mortier
Jul 17 at 16:07