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Question About Textbook Explanation of Normalized Property of Probability Mass Functions on Discrete Random Variables
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My textbook says the following when describing the conditions of a function to be a probability mass function on a random variable $mathbbx$:
$sum_x in mathbbx P(x) = 1$. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring.
Shouldn't it say "of many events occurring" instead of "of one of many events occurring"? The latter wouldn't necessarily be true, and it also seems out of context, since we're talking about the summation over all $x$ in the support?
probability-theory random-variables
asked Jul 14 at 14:57
Wyuw
1418
add a comment |Â
up vote
0
down vote
favorite
My textbook says the following when describing the conditions of a function to be a probability mass function on a random variable $mathbbx$:
$sum_x in mathbbx P(x) = 1$. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring.
Shouldn't it say "of many events occurring" instead of "of one of many events occurring"? The latter wouldn't necessarily be true, and it also seems out of context, since we're talking about the summation over all $x$ in the support?
probability-theory random-variables
asked Jul 14 at 14:57
Wyuw
1418
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My textbook says the following when describing the conditions of a function to be a probability mass function on a random variable $mathbbx$:
$sum_x in mathbbx P(x) = 1$. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring.
Shouldn't it say "of many events occurring" instead of "of one of many events occurring"? The latter wouldn't necessarily be true, and it also seems out of context, since we're talking about the summation over all $x$ in the support?
probability-theory random-variables
asked Jul 14 at 14:57
Wyuw
1418
My textbook says the following when describing the conditions of a function to be a probability mass function on a random variable $mathbbx$:
$sum_x in mathbbx P(x) = 1$. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring.
Shouldn't it say "of many events occurring" instead of "of one of many events occurring"? The latter wouldn't necessarily be true, and it also seems out of context, since we're talking about the summation over all $x$ in the support?
probability-theory random-variables
asked Jul 14 at 14:57
Wyuw
1418
edited Jul 14 at 15:02
asked Jul 14 at 14:57
Wyuw
1418
asked Jul 14 at 14:57
Wyuw
1418
asked Jul 14 at 14:57
Wyuw
1418
1418
add a comment |Â
add a comment |Â
1 Answer
1
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Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^prime$, by $P^prime(x) = P(x) / sum_y P(y)$.
answered Jul 14 at 15:19
parsiad
16k32253
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^prime$, by $P^prime(x) = P(x) / sum_y P(y)$.
answered Jul 14 at 15:19
parsiad
16k32253
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
add a comment |Â
up vote
1
down vote
accepted
Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^prime$, by $P^prime(x) = P(x) / sum_y P(y)$.
answered Jul 14 at 15:19
parsiad
16k32253
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^prime$, by $P^prime(x) = P(x) / sum_y P(y)$.
answered Jul 14 at 15:19
parsiad
16k32253
Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^prime$, by $P^prime(x) = P(x) / sum_y P(y)$.
answered Jul 14 at 15:19
parsiad
16k32253
answered Jul 14 at 15:19
parsiad
16k32253
answered Jul 14 at 15:19
parsiad
16k32253
answered Jul 14 at 15:19
parsiad
16k32253
16k32253
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
add a comment |Â
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Ok, thanks for the help.
– Wyuw
Jul 14 at 15:25
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
Out of curiosity, what is $sum_y P(y)$ supposed to be?
– Wyuw
Jul 14 at 15:26
1
1
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
It's the same as $sum_x P(x)$, I just used a different variable name.
– parsiad
Jul 14 at 15:30
add a comment |Â
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