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Finding the mean and variance of a probability distribution

posted Mar 17, 2014, 6:22 PM by Prof Kiernan
To find the mean of a probability distribution you need to use this formula:

In English this means you need to multiply each x value by the corresponding probability and get the sum of the results.


To find the variance of a probability distribution you need to use this formula:
In English this means you need to do the following:
First: subtract the mean from each x value and square each answer
Second: Multiply each answer from the first step by each probability
Third: get the sum of the answers from the second step

Note: remember the standard deviation is the square root of the variance.

Example 1: Find the mean and variance of the following probability distribution.

 x P(x)
 0  0.1
 1  0.3
 2  0.4
 3  0.2

To find the mean of the distribution we need to add another vertical column onto our table and a total row at the bottom of our table. and compute the x*P(x) for each x value then get the total of the column as our mean.

 x P(x)
x*P(x)
 0  0.1  0*0.1=0
 1  0.3  1*0.3=0.3
 2  0.4  2*0.4=0.8
 3  0.2  3*0.2=0.6
 Total     1.7  

Thus the mean of example 1 is  1.7 .

To find the variance of the distribution we need to add 3 new vertical columns onto our original table and a total row at the bottom of the table. The computations in each of the new columns are as follows:
In the first new column, subtract the mean from each x value (remember our mean is  1.7 )
 x  P(x)  x-mean    
 0  0.1  0 -  1.7  = -1.7
   
 1  0.3  1 -  1.7  = -0.7    
 2  0.4  2 -  1.7  = 0.3    
 3  0.2  3 -  1.7  = 1.3    
 Total        

In the second new column, square each answer from the first new column.
 x P(x) x-mean (x-mean)2 
 0 0.1 0 -  1.7  = -1.7
 (-1.7)2 = 2.89
 
 1 0.3 1 -  1.7  = -0.7  (-0.7)2 =0.49 
 2 0.4 2 -  1.7  = 0.3  0.32 =0.09 
 3 0.2 3 -  1.7  = 1.3  1.32 =1.69 
 Total    

In the third new column, multiply each answer from the second new column by each probability and finally get the sum of the answers from this step.
 x P(x) x-mean (x-mean)2 (x-mean)2*P(x)
 0 0.1 0 -  1.7  = -1.7
 (-1.7)2 = 2.89
 2.89*0.1 = 0.289
 1 0.3 1 -  1.7  = -0.7  (-0.7)2 =0.49 0.49*0.3 =0.147
 2 0.4 2 -  1.7  = 0.3  0.32 =0.09 0.09*0.4 =0.036
 3 0.2 3 -  1.7  = 1.3  1.32 =1.69 1.69*0.2 = 0.338
 Total    0.81

Thus our variance for example 1 is 0.81. Which means our standard deviation for this example is the square root of our variance or:

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