Question 1202695
<font color=black size=3>
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td></tr><tr><td>-1</td><td>2c</td></tr><tr><td>1</td><td>3c</td></tr><tr><td>2</td><td>4c</td></tr></table>


To have a valid probability distribution, the p(x) values must add to 1.
2c+3c+4c = 1
9c = 1
<font color=red>c = 1/9</font>


So,
2c = 2*(1/9) = 2/9
3c = 3*(1/9) = 3/9 = 1/3
4c = 4*(1/9) = 4/9


We get this updated table.
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td></tr><tr><td>-1</td><td>2/9</td></tr><tr><td>1</td><td>1/3</td></tr><tr><td>2</td><td>4/9</td></tr></table>
We see that P(x=-1) = <font color=red>2/9</font>
2/9 = 0.222 approximately


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This video goes over an example how to calculate the standard deviation of a probability distribution.
<a href="https://www.youtube.com/watch?v=YMbt5rYzp-Q">https://www.youtube.com/watch?v=YMbt5rYzp-Q</a>


The professor mentions these two formulas
{{{sigma = sqrt( sum((x-mu)^2*P(x),"","") )}}}
and
{{{sigma = sqrt( (sum(x^2*P(x),"","")) - mu^2 )}}}


where,
mu = {{{mu}}} = greek letter to represent the mean
lowercase sigma = {{{sigma}}} = greek letter for the standard deviation


The fancy looking "E" represents a summation. It's the greek uppercase letter sigma.
Something like {{{sum(x^2*P(x))}}} tells us to add up all of the x^2*P(x) terms.


Both formulas involve mu, so let's determine that value first.


mu = {{{sum(x*P(x)^"")}}}
mu = add up the x*P(x) terms
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td><td>x*P(x)</td></tr><tr><td>-1</td><td>2/9</td><td>-2/9</td></tr><tr><td>1</td><td>1/3</td><td>1/3</td></tr><tr><td>2</td><td>4/9</td><td>8/9</td></tr></table>
The sum of the x*P(x) values is:
-2/9 + 1/3 + 8/9
= -2/9 + 3/9 + 8/9
= (-2 + 3 + 8)/9
= 9/9
= 1


Therefore, mu = 1.



Let's say we used the formula
{{{sigma = sqrt( sum((x-mu)^2*P(x),"","") )}}}
What we'll need to do is<ol><li>Subtract mu from each x value to get x-mu.</li><li>Square the difference to get (x-mu)^2</li><li>Multiply those squares with P(x) to get (x-mu)^2*P(x).</li></ol>This is what your table should look like
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td><td>x-mu</td><td>(x-mu)^2</td><td>(x-mu)^2*P(x)</td></tr><tr><td>-1</td><td>2/9</td><td>-2</td><td>4</td><td>8/9</td></tr><tr><td>1</td><td>1/3</td><td>0</td><td>0</td><td>0</td></tr><tr><td>2</td><td>4/9</td><td>1</td><td>1</td><td>4/9</td></tr></table>
Add up the values in the (x-mu)^2*P(x) column.
8/9 + 0 + 4/9 = 12/9 = 4/3
That's the variance, so the standard deviation is the square root of that
{{{sqrt(4/3) = sqrt(4)/sqrt(3)}}} 


{{{sqrt(4/3) = 2/sqrt(3)}}}


{{{sqrt(4/3) = (2*sqrt(3))/(sqrt(3)*sqrt(3))}}} 


{{{sqrt(4/3) = (2*sqrt(3))/(3)}}} 


{{{sqrt(4/3) = (2/3)*sqrt(3)}}} 
The standard deviation is exactly <font color=red>(2/3)*sqrt(3)</font>
When using a calculator, (2/3)*sqrt(3) = 1.15470053837926 approximately.


If you wanted to use the formula {{{sigma = sqrt( (sum(x^2*P(x),"","")) - mu^2 )}}}, then this is what your table could look like
<table border = "1" cellpadding = "5"><tr><td>x</td><td>P(x)</td><td>x*P(x)</td><td>x^2</td><td>x^2*P(x)</td></tr><tr><td>-1</td><td>2/9</td><td>-2/9</td><td>1</td><td>2/9</td></tr><tr><td>1</td><td>1/3</td><td>1/3</td><td>1</td><td>1/3</td></tr><tr><td>2</td><td>4/9</td><td>8/9</td><td>4</td><td>16/9</td></tr></table>
Add up the x^2*P(x) values
{{{sum(x^2*P(x)) = 2/9 + 1/3 + 16/9}}}


{{{sum(x^2*P(x)) = 2/9 + 3/9 + 16/9}}}


{{{sum(x^2*P(x)) = 21/9}}}


{{{sum(x^2*P(x)) = 7/3}}}


Then plug that result into the formula I mentioned earlier
{{{sigma = sqrt( (sum(x^2*P(x),"","")) - mu^2 )}}}


{{{sigma = sqrt( 7/3 - 1^2 )}}}


{{{sigma = sqrt( 7/3 - 1 )}}}


{{{sigma = sqrt( 7/3 - 3/3 )}}}


{{{sigma = sqrt( 4/3 )}}}


{{{sigma = 2/sqrt(3)}}}


{{{sigma = (2*sqrt(3))/(3)}}} 


{{{sigma = (2/3)*sqrt(3)}}} 


This calculator can be used to check your work
<a href="https://www.mathportal.org/calculators/statistics-calculator/probability-distributions-calculator.php">https://www.mathportal.org/calculators/statistics-calculator/probability-distributions-calculator.php</a>


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Summary: 


<font color=red>c = 1/9</font>
P(x = -1) = <font color=red>2/9</font>
Standard deviation = <font color=red>(2/3)*sqrt(3)</font> = 1.15470053837926 approximately
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