More Parameters

There are several other parameters that we need to mention. The first is the probability model variance.  To understand this parameter, return to spinner example  of the last section. Recall we had spun a fair spinner 100 times. This resulted in our sample. Recall that we computed the sample mean and this motivated the probability model mean $\mu$. Suppose next we calculate the sample variance s². But we can do this easy using the tallied sample results as follows:

$$s^2 = \bigg( (1 - 1.83)^2 \times \frac{43}{100} \bigg) + \big... ...igg) + \bigg( (3-1.83)^2 \times \frac{26}{100} \bigg) = .6611$$

Actually I should have divided by 99 instead of 100, but with n so large it won't matter much. By the last line it is easy to see what s² is estimating. Again, .43 is our estimate of p(1), .31 is our estimate of p(2), .26 is our estimate of p(3), and 1.83 estimates $\mu$. Hence, s² is estimating

$$\sigma^2 = \big((1-2)^2 \times p(1) \big) + \big( (2-2)^2 \times p(2) \big) + \big( (3-2)^2 \times p(3) \big)$$

That's the Greek letter sigma . $\sigma^2$  is called the probability model variance  and its square root $\sigma$ is called the probability model standard deviation . It is the center of gravity along the horizontal axis of the graph of the probability model. So in this example, assuming the spinner is fair, .6611 is an estimate of $\sigma^2 = 2/3$ and its square root, .8131, is an estimate of $\sigma = \sqrt{2/3} = .8165$.
Three other parameters of interest are the median and quartiles of the probability model. These are used more for the continuous probability models, so we will present them later.

Exercise 5.4.1  

1.

Let S denote the sum of two numbers spun on a fair spinner with the numbers 1, 2, and 3 on it. The range of S is 2, 3, 4, 5, 6. Determine the probability model variance of S.

2.

Let X denote the number of aces in a 2 card hand drawn without replacement from a well shuffled standard deck 0f 52 cards. Then the range of X is {0, 1, 2}. Determine the probability model variance of X.

3.

Repeat the last problem under sampling with replacement.

4.

In the urn problem (with the balls well mixed ) discussed above, let Z denote the number of red balls in the sample (without replacement) of size 3. Determine the probability model variance of Z.