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Guassian Probability density function

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data

    The PDF (probability density function) of a Gaussian variable x is given by.

    $$p_x(x)=\frac{1}{C \sqrt{2 \pi}} e^{\frac{-(x-4)^2}{18}}$$

    a) Find C
    b)find the probability of x≥2 --> ##P(x≥2)##

    2. Relevant equations

    $$ \frac{dF_X(x)}{dx} x=P(x<X≤x+Δx)$$

    3. The attempt at a solution

    So i get stuck on how to solve the above for C. I have an example of a similar problem that my professor did in class but it skips a lot of steps that I need to see to fully understand. It seems like he started with taking the integral of the signal by using an integral table?

    In my text book I do see that the above is a standard of a gaussian or normal probability density. It looks someting like this.

    $$p_X(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^2}{2}$$
    $$F_X(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} e^{\frac{-x^2}{2}}dx$$

    Any hints on where to start?

    Any help is much appreciated! Thank you!
     
  2. jcsd
  3. Mar 31, 2013 #2
    The parameter C is the standard deviation.
    The denominator of the power of the exponent is equal to 2(C^2).
    Hence C = 3, as 2 times 9 is 18.

    [tex]p_x(x)=\frac{1}{3 \sqrt{2\pi}}e^{\frac{-(x-4)^2}{18}}[/tex]

    My approach to this was to look at the formula given on this wikipedia page:
    https://en.wikipedia.org/wiki/Normal_distribution

    To find the probability of x=2 I think maybe we could substitute.

    [tex]p_x(2)=\frac{1}{3 \sqrt{2\pi}}e^{\frac{-(2-4)^2}{18}}[/tex]
     
    Last edited: Mar 31, 2013
  4. Mar 31, 2013 #3
    [tex]x^2\sqrt{x}[\tex]
    was trying some latex here. semi-success.
     
  5. Mar 31, 2013 #4
    Substituted the value for x=2 in my Casio and I get 0.1064826685.
    Will try and plot in Mathematica for confirmation.
     
  6. Mar 31, 2013 #5
    Have plotted them :)

    The files are attached to this post. Need to work out how to get them to flash up here.
     

    Attached Files:

  7. Mar 31, 2013 #6
    Plotted :)

    Not sure quite how to embed the image so that it appears in this post. But its attached & the hand-calculated value looks reasonable :)

    GaussianPhysForum.gif

    GaussianPhysForumzoom.gif

    yes! Think this is how its done.
     
  8. Mar 31, 2013 #7
    Wow it was really that simple! I had that equation written down right on the scratch pad where I was working this problem and didnt see that I guess.

    To find the probability I followed that other example that simply used the ##Q(x)## function. And then take the result and look up the probability in the table that goes along with that function.

    For reference the function looks like this (from my text) ## Q(x)= \frac{1}{x \sqrt{2 \pi}} e^{\frac{-x^2}{2}}##

    Thanks for the help AugustCrawl!
     
  9. Mar 31, 2013 #8
    Oh btw a little note. Your latex code looks ok. If you use those tage be sure to use "[\itex]" i think your just leaving out the i. Or you can use two dollar signs $$ before and after for a separate line of code or two hash tags ## for code to be on the same line.
     
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