If P(X < a) = b then is P(f(X) < f(a)) = b?
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If f(x) is a monotonic increasing function, then does P(X < a) = b imply P(f(x) < f(a)) = b ? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
mathematical-statistics function measure-theory
asked 1 hour ago
Linsu Han
62
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up vote
1
down vote
favorite
If f(x) is a monotonic increasing function, then does P(X < a) = b imply P(f(x) < f(a)) = b ? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
mathematical-statistics function measure-theory
asked 1 hour ago
Linsu Han
62
By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If f(x) is a monotonic increasing function, then does P(X < a) = b imply P(f(x) < f(a)) = b ? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
mathematical-statistics function measure-theory
asked 1 hour ago
Linsu Han
62
If f(x) is a monotonic increasing function, then does P(X < a) = b imply P(f(x) < f(a)) = b ? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
mathematical-statistics function measure-theory
mathematical-statistics function measure-theory
asked 1 hour ago
Linsu Han
62
asked 1 hour ago
Linsu Han
62
asked 1 hour ago
Linsu Han
62
asked 1 hour ago
Linsu Han
62
asked 1 hour ago
Linsu Han
62
62
By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago
add a comment |
By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago
By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago
By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago
add a comment |
1 Answer
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Consider the set of $x$, call it $S$, where $x<a$. You seek for the probability, $P(S)$. Any expression that lead to $S$ produces the exact same probability, $b$, irrespective of its decleration. If $f(x)$ is a monotonic (strictly) increasing function, $x<a$ directly implies $f(x)<f(a)$ and vice versa, i.e. if $f(x)<f(a)$, then $x<a$.
answered 39 mins ago
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Consider the set of $x$, call it $S$, where $x<a$. You seek for the probability, $P(S)$. Any expression that lead to $S$ produces the exact same probability, $b$, irrespective of its decleration. If $f(x)$ is a monotonic (strictly) increasing function, $x<a$ directly implies $f(x)<f(a)$ and vice versa, i.e. if $f(x)<f(a)$, then $x<a$.
answered 39 mins ago
gunes
2,434111
add a comment |
up vote
3
down vote
Consider the set of $x$, call it $S$, where $x<a$. You seek for the probability, $P(S)$. Any expression that lead to $S$ produces the exact same probability, $b$, irrespective of its decleration. If $f(x)$ is a monotonic (strictly) increasing function, $x<a$ directly implies $f(x)<f(a)$ and vice versa, i.e. if $f(x)<f(a)$, then $x<a$.
answered 39 mins ago
gunes
2,434111
add a comment |
up vote
3
down vote
up vote
3
down vote
Consider the set of $x$, call it $S$, where $x<a$. You seek for the probability, $P(S)$. Any expression that lead to $S$ produces the exact same probability, $b$, irrespective of its decleration. If $f(x)$ is a monotonic (strictly) increasing function, $x<a$ directly implies $f(x)<f(a)$ and vice versa, i.e. if $f(x)<f(a)$, then $x<a$.
answered 39 mins ago
gunes
2,434111
Consider the set of $x$, call it $S$, where $x<a$. You seek for the probability, $P(S)$. Any expression that lead to $S$ produces the exact same probability, $b$, irrespective of its decleration. If $f(x)$ is a monotonic (strictly) increasing function, $x<a$ directly implies $f(x)<f(a)$ and vice versa, i.e. if $f(x)<f(a)$, then $x<a$.
answered 39 mins ago
gunes
2,434111
answered 39 mins ago
gunes
2,434111
answered 39 mins ago
gunes
2,434111
answered 39 mins ago
gunes
2,434111
2,434111
add a comment |
add a comment |
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By "monotonic increasing" do you mean strictly increasing (if $a < b$ then $f(a) < f(b)$) or non-decreasing (if $a < b$ then $f(a) leq f(b)$)?
– Artem Mavrin
1 hour ago