Question 1206570
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𝜇 = mu
𝜎 = sigma


Given table
<table border = "1" cellpadding = "5"><tr><td>x</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>p(x)</td><td>0.25</td><td>0.05</td><td>0.15</td><td>0.2</td><td>0.05</td><td>0.3</td></tr></table>


Introduce a new row called x*p(x), where you multiply the paired x and p(x) values.
Spreadsheet software is recommended.
<table border = "1" cellpadding = "5"><tr><td>x</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>p(x)</td><td>0.25</td><td>0.05</td><td>0.15</td><td>0.2</td><td>0.05</td><td>0.3</td></tr><tr><td>x*p(x)</td><td>0</td><td>0.05</td><td>0.3</td><td>0.6</td><td>0.2</td><td>1.5</td></tr></table>
Add up the x*p(x) values to get the expected value.
mu = mean = expected value = E[X]
mu = sum of the x*p(x) values
mu = 0+0.05+0.3+0.6+0.2+1.5 
mu = <font color=red>2.65</font>



We'll use that value of mu to determine the variance, and by extension, the standard deviation as well.


Introduce a new row called (x-mu)^2*p(x)
The naming should be self-explanatory. If not then please let me know.
Example calculation: if x = 0, then (x-mu)^2*p(x) = (0-2.65)^2*0.25 = 1.755625
<table border = "1" cellpadding = "5"><tr><td>x</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>p(x)</td><td>0.25</td><td>0.05</td><td>0.15</td><td>0.2</td><td>0.05</td><td>0.3</td></tr><tr><td>x*p(x)</td><td>0</td><td>0.05</td><td>0.3</td><td>0.6</td><td>0.2</td><td>1.5</td></tr><tr><td>(x-mu)^2*p(x)</td><td>1.755625</td><td>0.136125</td><td>0.063375</td><td>0.0245</td><td>0.091125</td><td>1.65675</td></tr></table>
sigma^2 = variance
sigma^2 = sum of the (x-mu)^2*p(x) values
sigma^2 = 1.755625+0.136125+0.063375+0.0245+0.091125+1.65675 
sigma^2 = <font color=red>3.7275</font>


Side note: another way to find the variance is to compute E[X^2] - (E[X])^2 aka E[X^2] - mu^2
I'll leave this as an exercise to the reader.


Then,
sigma = standard deviation
sigma = sqrt( variance )
sigma = sqrt( 3.7275 )
sigma = 1.9306735 approximately
When rounding to 2 decimal places we get <font color=red>1.93</font>


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Answers:
mu = <font color=red>2.65</font>
sigma^2 = <font color=red>3.7275</font>
sigma = <font color=red>1.93</font>
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