Guassian Probability density function

[Evo8]

Homework Statement

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 --> $P(x≥2)$

Homework Equations

$$ \frac{dF_X(x)}{dx} x=P(x<X≤x+Δx)$$

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 textbook I do see that the above is a standard of a gaussian or normal probability density. It looks something 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!

 

[AugustCrawl]

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.

$$p_x(x)=\frac{1}{3 \sqrt{2\pi}}e^{\frac{-(x-4)^2}{18}}$$

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.

$$p_x(2)=\frac{1}{3 \sqrt{2\pi}}e^{\frac{-(2-4)^2}{18}}$$

 

[AugustCrawl]

$$x^2\sqrt{x}[\tex]<br /> was trying some latex here. semi-success.$$

 

[AugustCrawl]

Substituted the value for x=2 in my Casio and I get 0.1064826685.
Will try and plot in Mathematica for confirmation.

 

[AugustCrawl]

Have plotted them :)

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

 

[AugustCrawl]

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 :)

yes! Think this is how its done.

 

[Evo8]

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!

 

[Evo8]

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.