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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 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: