We can check for the first question "Is it a probability distribution? " by adding all the values on the right hand side column and confirming that they add to 1 (one) which is the same as 100 percent.
0.02 + 0.15 + 0.29 + 0.31 + 0.23 = 1.0
So,this seems to be a probability distribution.
Then they want to know what the mean of the distribution is, that is what is the average for "x" which is the number of households that have a child with dissability. So we do the addition of all of the events by multiplying each x times the associated probability and we then get:
(0 * 0.02 + 1 * 0.15 + 2 * 0.29 + 3 * 0.31 + 4 * 0.23)/10 = 0.258
So the mean is: 0.258. (equivalent to 25.8 %)
Now we want to calculate the standard deviation for this.
The standard deviation involves first calculatting the deviations of each event minus the average (mean) that we just calculated, squaring these values, adding the squares, dividing the total by the number of events minus one and finally finding the square root of the value ontained.
I am going to use a program called Excel to produc the results much faster if I do it by hand.
The result came out as 0.052662 (equivalent to about 5.3%)
so we have that the average (mean) is 25.8 percent and the standard deviation is about 5.3 %