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







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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
    54 mins ago















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







share|cite|improve this question






















  • 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
    54 mins ago













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







share|cite|improve this question













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






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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
    54 mins 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
    54 mins 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
54 mins 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
54 mins ago










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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 $x<a$.






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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 $x<a$.






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      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 $x<a$.






      share|cite|improve this answer























        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 $x<a$.






        share|cite|improve this answer












        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 $x<a$.







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        answered 32 mins ago









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