How to Find the Standard Deviation of a Discrete Probability Distribution

Discrete Probability Distributions

Some of this lecture should be very familiar to you, just old topics with new words attached to them. Other parts will expand the concepts we've been developing.

Random Variable

The first concept is the random variable. This just means assigning a number to each outcome of a probability experiment. We've already done this for rolling two dice: the sum of the upward-facing pips is the random variable. Flipping four coins: the number of heads is the random variable.

But there are two kinds of random variables, discrete and continuous. We used these words in the very first lecture in the class. Discrete random variables take on counting numbers as values: 0, 1, 2, and so on. Maybe there's a definite stopping place, like 12 for rolling two dice, or 4 for the number of heads in flipping four coins, but sometimes there isn't, or we don't know what it is. For instance, if the random variable is the number of phone calls a business receives in a given hour of the day, we don't know the highest possible value of the random variable, but we know the value will be 0, 1, 2, or a larger whole number.

Today we're going to talk only about discrete random variables and their probability distributions. We'll take on the continuous ones, like giving the height of a person when the probability experiment is to measure him or her, in a later chapter.

A Probability Distribution from Classical Probability

What if you're going to have four children and each one will be either or boy or a girl, and the likelihood of either sex is the same for each birth? We can say that the sex of each child is independent of the sex of the others. This is the same as flipping four fair coins. (Some people don't believe any of this. They think that some people are more likely to produce boys than girls, or vice versa, or that once having produced a child of one sex you go on producing this sex. I'm not disputing these beliefs, but to have an example from classical probability we can't employ them.) Here's the sample space, using F for a girl and M for a boy, and listing the sexes in the order of birth:

FFMM

FMFM

FMMF

MFFM

MFMF

MMFF

The size of the sample space (n(S)) is 16, as you know from the multiplication rule of counting: . I've put boxes around the outcomes which result in the same number of boys, which is what we're going to use for the random variable, which we will call X. As you can see, there is one outcome resulting in 0 boys, four resulting in 1 boy, six resulting in 2 boys, four resulting in 3 boys, and one resulting in 4 boys.

Using the assumption of classical probability, that any outcome is as likely to occur as any other outcome, we can use the formula , where the event E is that you have X boys, to determine the probability that you have 0, 1, 2, 3, or 4 boys. So instead of saying "the probability of having 1 boy", we go to the much more compact notation of . Since  ranges from 0 to 4, we can put all the probabilities into a table which is called a probability distribution:

x	P(X)
0	1/16
1	4/16
2	6/16
3	4/16
4	1/16

The left-hand column lists all possible values of the random variable X, and the right-hand column lists the probability that the value X will occur. So P(0) = 0.0625, P(1) = 0.25, etc. The P(X)'s have to add up to one, since one of the values of X has to occur each time the experiment (in this case, having four children) is performed. Another requirement for a discrete probability distribution is for each  to be between 0 and 1, otherwise they don't make sense as probabilities.

Mean/Expected Value of a Discrete Distribution

As with any data set, we want to know two things: a measure of central tendency and a measure of variation. In a probability distribution, this will be the population mean, μ, and the population standard deviation, σ. Note that these are parameters, numbers describing a population, because the probability distribution describes the total behavior of the random variable, not just a sample of it.

Here's how you find μ: first you multiply each value of the random variable by its associated probability. Since 0*(1/16)=0, you get 0 for the first entry. The second is 1*(4/16)=0.25, and so on. Here's the completed column added on:

x	P(X)
0	1/16	0
1	4/16	0.25
2	6/16	0.75
3	4/16	0.75
4	1/16	0.25

Now you add up the last column, whose sum can be expressed as , using our symbol for the sum. As you can see, the sum is 0+0.25+0.75+0.75+0.25 = 2, and this is μ, the population mean. So the formula for the population mean is , and it is our measure of central tendency.

Another name for this μ is the expected value (represented by E(X)), because it's the value of the random variable which you would most expect to find over many repeated experiments. In the case of having four children, you would expect on average that half, or 2, would be boys. But it's neat to see that it works out this way also by adding up the  values. It works here because the probabilities are symmetrical around    in other words P(0) = P(4) and P(1) = P(3).

In general expected value is just a synonym for the mean, but it's a useful concept for answering questions like how many boys one would expect to find in 20 families with four children. Since you expect 2 per family, the answer is 20*2=40. Notice that it's a coincidence that we happen to have a whole number as the expected value -- to see this, simply change the number of kids from 4 to 3, and you will find that . You should avoid rounding an expected value to a whole number for a discrete random variable -- E(X) does NOT appear as an outcome for the random variable, but is the weighted mean of all the outcomes by taking into account the probability of each outcome.

Standard Deviation of a Discrete Distribution

How about σ? This is a somewhat more complicated calculation, but it's just an extension of the work we have done to find the expected value. First, you find x-μ, the deviation from the mean, for each value of X. For 0, this would be 0-2=-2, for 1 it would be 1-2=-1, and so on:

x	P(X)
0	1/16	0	-2
1	4/16	0.25	-1
2	6/16	0.75	0
3	4/16	0.75	1
4	1/16	0.25	2
		=2

This is somewhat reminiscent of how we found the standard deviation of a data set using the table method, and so is the next step, in which we square the (x-μ)'s, remembering that squares are never negative:

x	P(X)
0	1/16	0.0625	-2	4
1	4/16	0.25	-1	1
2	6/16	0.75	0	0
3	4/16	0.75	1	1
4	1/16	0.25	2	4
		=2

Similar to the earlier work we have done on finding the standard deviation from frequency distributions; now we have to take into account the different probabilities, or weights, of the different values of X. What we do is multiply the 's by the probability of the X's that produced them. The first one is 0.0625*4=0.25, the next 0.25*1=0.25 , and so on (be careful to multiply by P(X) and not x*P(X)):

x	P(X)
0	1/16	0	-2	4	0.25
1	4/16	0.25	-1	1	0.25
2	6/16	0.75	0	0	0
3	4/16	0.75	1	1	0.25
4	1/16	0.25	2	4	0.25
		=2.0			=1.0

The symmetry we observed earlier appeared again here in the last column.

Finally we've finished the table, and now we do something that is again reminiscent of the table method for finding the standard deviation of a data set: add up the last column: . However, we don't divide by n-1 this time: this sum is the population variance,  (remember that in the corresponding process for data sets, first we got , the sample variance).

So the formula is , and from this we get the formula for the population standard deviation,  . So in the case of the four children, , or 1.0. This was a special case  the standard deviation is generally not a whole number, since it's coming from the square root of variance. So rounding it to the 1st decimal is the most likely solution for most discrete distributions.

How to Find the Standard Deviation of a Discrete Probability Distribution

Source: http://www.santarosa.edu/~ylin/Math15/notes/Discrete_Probability_Distributions.htm

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