Normal Distribution & Modulus











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I was doing some normal distribution questions and was stumped when I faced this question. Hope you guys can give me a hand here.



Question
If the annual precipitation $X$ in a city is a normal variable, with mean $50$cm and standard deviation $10$cm. Determine the following.
The probability that $X$ is within $5$cm from the mean annual precipitation



The answer
The probability that $X$ is within $5$cm from the mean annual precipitation is
$P(|X - 50| < 5) = P(|(X-50/10)| < 5/10).$



My Confusion
Why is $P(|(X-50/10)| < 5/10)$ not P(|(X-50**-50**/10)| < 5/10). And why divide the $5$ with $10?$



Thank you guys










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  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Nov 17 at 7:32










  • Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
    – BruceET
    Nov 17 at 21:19















up vote
1
down vote

favorite












I was doing some normal distribution questions and was stumped when I faced this question. Hope you guys can give me a hand here.



Question
If the annual precipitation $X$ in a city is a normal variable, with mean $50$cm and standard deviation $10$cm. Determine the following.
The probability that $X$ is within $5$cm from the mean annual precipitation



The answer
The probability that $X$ is within $5$cm from the mean annual precipitation is
$P(|X - 50| < 5) = P(|(X-50/10)| < 5/10).$



My Confusion
Why is $P(|(X-50/10)| < 5/10)$ not P(|(X-50**-50**/10)| < 5/10). And why divide the $5$ with $10?$



Thank you guys










share|cite|improve this question
























  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Nov 17 at 7:32










  • Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
    – BruceET
    Nov 17 at 21:19













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I was doing some normal distribution questions and was stumped when I faced this question. Hope you guys can give me a hand here.



Question
If the annual precipitation $X$ in a city is a normal variable, with mean $50$cm and standard deviation $10$cm. Determine the following.
The probability that $X$ is within $5$cm from the mean annual precipitation



The answer
The probability that $X$ is within $5$cm from the mean annual precipitation is
$P(|X - 50| < 5) = P(|(X-50/10)| < 5/10).$



My Confusion
Why is $P(|(X-50/10)| < 5/10)$ not P(|(X-50**-50**/10)| < 5/10). And why divide the $5$ with $10?$



Thank you guys










share|cite|improve this question















I was doing some normal distribution questions and was stumped when I faced this question. Hope you guys can give me a hand here.



Question
If the annual precipitation $X$ in a city is a normal variable, with mean $50$cm and standard deviation $10$cm. Determine the following.
The probability that $X$ is within $5$cm from the mean annual precipitation



The answer
The probability that $X$ is within $5$cm from the mean annual precipitation is
$P(|X - 50| < 5) = P(|(X-50/10)| < 5/10).$



My Confusion
Why is $P(|(X-50/10)| < 5/10)$ not P(|(X-50**-50**/10)| < 5/10). And why divide the $5$ with $10?$



Thank you guys







statistics probability-distributions






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edited Nov 17 at 21:22









BruceET

34.8k71440




34.8k71440










asked Nov 17 at 7:27









BEARBEARTHEBEAR

61




61












  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Nov 17 at 7:32










  • Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
    – BruceET
    Nov 17 at 21:19


















  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Nov 17 at 7:32










  • Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
    – BruceET
    Nov 17 at 21:19
















Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 17 at 7:32




Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 17 at 7:32












Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
– BruceET
Nov 17 at 21:19




Tried to edit your Question, but couldn't figure out the **'s, so maybe you can clarify that part. Hope one of the Answers is helpful.
– BruceET
Nov 17 at 21:19










2 Answers
2






active

oldest

votes

















up vote
1
down vote













We know that $frac{X-mu}{sigma} sim N(0,1)$, hence we divide by the standard deviation.



When we have an inequality $a le b$ we can divide both sides by the same positive scalar and the statements are equivalent.






share|cite|improve this answer

















  • 1




    Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
    – BruceET
    Nov 17 at 20:30


















up vote
1
down vote













If $X sim mathsf{Norm}(mu = 50, sigma=10),$ then



$$P(45 < X < 55) = Pleft(frac{45 - 50}{10}< frac{X-mu}{sigma} < frac{55 - 50}{10}right)\
= P(-0.5 < Z < 0.5) = 0.3829,$$

where $Z sim mathsf{Norm}(0, 1),$ the standard normal distribution.



The displayed equation shows the process of 'standardization', which makes it possible to find the (approximate) probability from printed tables of the standard normal distribution. For example, you can find from such tables that
$P(0 < Z < 0.5) = 0.1915,$ from which the result can be obtained by symmetry as
$P(-0.5 < Z < 0.5) = 2(0.1915) = 0.3830,$ correct to three places.



Alternatively, statistical software or statistical calculators can be used to evaluate the original expression $P(45 < X < 55) = 0.3829249$ directly. The result from
R statistical software is shown below.



diff(pnorm(c(45,55), 50, 10))
[1] 0.3829249


The left panel below shows the density function of $mathsf{Norm}(50, 10)$ and the panel at right shows the standard normal density function. In both panels, the total area beneath the density function is $1$ and the desired probability is
the area between the vertical dotted lines.



enter image description here






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    2 Answers
    2






    active

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

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    active

    oldest

    votes








    up vote
    1
    down vote













    We know that $frac{X-mu}{sigma} sim N(0,1)$, hence we divide by the standard deviation.



    When we have an inequality $a le b$ we can divide both sides by the same positive scalar and the statements are equivalent.






    share|cite|improve this answer

















    • 1




      Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
      – BruceET
      Nov 17 at 20:30















    up vote
    1
    down vote













    We know that $frac{X-mu}{sigma} sim N(0,1)$, hence we divide by the standard deviation.



    When we have an inequality $a le b$ we can divide both sides by the same positive scalar and the statements are equivalent.






    share|cite|improve this answer

















    • 1




      Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
      – BruceET
      Nov 17 at 20:30













    up vote
    1
    down vote










    up vote
    1
    down vote









    We know that $frac{X-mu}{sigma} sim N(0,1)$, hence we divide by the standard deviation.



    When we have an inequality $a le b$ we can divide both sides by the same positive scalar and the statements are equivalent.






    share|cite|improve this answer












    We know that $frac{X-mu}{sigma} sim N(0,1)$, hence we divide by the standard deviation.



    When we have an inequality $a le b$ we can divide both sides by the same positive scalar and the statements are equivalent.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 17 at 7:31









    Siong Thye Goh

    94.8k1462114




    94.8k1462114








    • 1




      Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
      – BruceET
      Nov 17 at 20:30














    • 1




      Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
      – BruceET
      Nov 17 at 20:30








    1




    1




    Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
    – BruceET
    Nov 17 at 20:30




    Fine answer (+1), but I wanted to give a little more detail about dividing by 10.
    – BruceET
    Nov 17 at 20:30










    up vote
    1
    down vote













    If $X sim mathsf{Norm}(mu = 50, sigma=10),$ then



    $$P(45 < X < 55) = Pleft(frac{45 - 50}{10}< frac{X-mu}{sigma} < frac{55 - 50}{10}right)\
    = P(-0.5 < Z < 0.5) = 0.3829,$$

    where $Z sim mathsf{Norm}(0, 1),$ the standard normal distribution.



    The displayed equation shows the process of 'standardization', which makes it possible to find the (approximate) probability from printed tables of the standard normal distribution. For example, you can find from such tables that
    $P(0 < Z < 0.5) = 0.1915,$ from which the result can be obtained by symmetry as
    $P(-0.5 < Z < 0.5) = 2(0.1915) = 0.3830,$ correct to three places.



    Alternatively, statistical software or statistical calculators can be used to evaluate the original expression $P(45 < X < 55) = 0.3829249$ directly. The result from
    R statistical software is shown below.



    diff(pnorm(c(45,55), 50, 10))
    [1] 0.3829249


    The left panel below shows the density function of $mathsf{Norm}(50, 10)$ and the panel at right shows the standard normal density function. In both panels, the total area beneath the density function is $1$ and the desired probability is
    the area between the vertical dotted lines.



    enter image description here






    share|cite|improve this answer



























      up vote
      1
      down vote













      If $X sim mathsf{Norm}(mu = 50, sigma=10),$ then



      $$P(45 < X < 55) = Pleft(frac{45 - 50}{10}< frac{X-mu}{sigma} < frac{55 - 50}{10}right)\
      = P(-0.5 < Z < 0.5) = 0.3829,$$

      where $Z sim mathsf{Norm}(0, 1),$ the standard normal distribution.



      The displayed equation shows the process of 'standardization', which makes it possible to find the (approximate) probability from printed tables of the standard normal distribution. For example, you can find from such tables that
      $P(0 < Z < 0.5) = 0.1915,$ from which the result can be obtained by symmetry as
      $P(-0.5 < Z < 0.5) = 2(0.1915) = 0.3830,$ correct to three places.



      Alternatively, statistical software or statistical calculators can be used to evaluate the original expression $P(45 < X < 55) = 0.3829249$ directly. The result from
      R statistical software is shown below.



      diff(pnorm(c(45,55), 50, 10))
      [1] 0.3829249


      The left panel below shows the density function of $mathsf{Norm}(50, 10)$ and the panel at right shows the standard normal density function. In both panels, the total area beneath the density function is $1$ and the desired probability is
      the area between the vertical dotted lines.



      enter image description here






      share|cite|improve this answer

























        up vote
        1
        down vote










        up vote
        1
        down vote









        If $X sim mathsf{Norm}(mu = 50, sigma=10),$ then



        $$P(45 < X < 55) = Pleft(frac{45 - 50}{10}< frac{X-mu}{sigma} < frac{55 - 50}{10}right)\
        = P(-0.5 < Z < 0.5) = 0.3829,$$

        where $Z sim mathsf{Norm}(0, 1),$ the standard normal distribution.



        The displayed equation shows the process of 'standardization', which makes it possible to find the (approximate) probability from printed tables of the standard normal distribution. For example, you can find from such tables that
        $P(0 < Z < 0.5) = 0.1915,$ from which the result can be obtained by symmetry as
        $P(-0.5 < Z < 0.5) = 2(0.1915) = 0.3830,$ correct to three places.



        Alternatively, statistical software or statistical calculators can be used to evaluate the original expression $P(45 < X < 55) = 0.3829249$ directly. The result from
        R statistical software is shown below.



        diff(pnorm(c(45,55), 50, 10))
        [1] 0.3829249


        The left panel below shows the density function of $mathsf{Norm}(50, 10)$ and the panel at right shows the standard normal density function. In both panels, the total area beneath the density function is $1$ and the desired probability is
        the area between the vertical dotted lines.



        enter image description here






        share|cite|improve this answer














        If $X sim mathsf{Norm}(mu = 50, sigma=10),$ then



        $$P(45 < X < 55) = Pleft(frac{45 - 50}{10}< frac{X-mu}{sigma} < frac{55 - 50}{10}right)\
        = P(-0.5 < Z < 0.5) = 0.3829,$$

        where $Z sim mathsf{Norm}(0, 1),$ the standard normal distribution.



        The displayed equation shows the process of 'standardization', which makes it possible to find the (approximate) probability from printed tables of the standard normal distribution. For example, you can find from such tables that
        $P(0 < Z < 0.5) = 0.1915,$ from which the result can be obtained by symmetry as
        $P(-0.5 < Z < 0.5) = 2(0.1915) = 0.3830,$ correct to three places.



        Alternatively, statistical software or statistical calculators can be used to evaluate the original expression $P(45 < X < 55) = 0.3829249$ directly. The result from
        R statistical software is shown below.



        diff(pnorm(c(45,55), 50, 10))
        [1] 0.3829249


        The left panel below shows the density function of $mathsf{Norm}(50, 10)$ and the panel at right shows the standard normal density function. In both panels, the total area beneath the density function is $1$ and the desired probability is
        the area between the vertical dotted lines.



        enter image description here







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 17 at 20:25

























        answered Nov 17 at 20:09









        BruceET

        34.8k71440




        34.8k71440






























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