Parameters

There are several parameters  associated with a continuous probability model that will prove useful. These are measures for probability models, so we will classify them into location and scale. To visualize these parameters, consider the probability model in Figure 5.6.

  

Figure 5.6: A continuous probability model

While reading the material below try to locate the parameters on it. Some of these will be "guesses". The answers will be given at the end.

Also consider the following random sample from this model, (I obtained this sample).

   24.0720    7.4773   18.4161   10.1440   10.4192    7.0660    3.9749 
    0.5231    7.9999   28.7453    6.0621   12.3729    5.3767   23.1134 
    7.9949   16.0966   10.4424    4.1915    8.4586    9.9292

As we proceed, calculate the statistics based on this sample which are estimates of the parameters. In fact, the stem leaf plot or the histogram are estimates of the probability curve. These estimates (stem leaf, histogram) are poor because the sample is so small.

1.

Location Parameters

The median  ,  $\theta$, is the point located along the horizontal axis of the probability model which divides the probability mass in two. That is, half the time the variable is less than $\theta$ and half the time the variable is greater than $\theta$. Okay! Locate the median on the probability model above.

Calculate the sample median, Q₂. This is the estimate of the median you located on the probability model above.

The mean , $\mu$, is the center of gravity along the horizontal axis of the probability model. This is the point where the probability mass would balance. Obviously if the probability curve is symmetric, the mean and median would agree . Locate the mean on the probability model above.

2.

Scale or Noise Parameters

The first quartile   , q₁, is the point located along the horizontal axis of the probability model which divides the probability mass into 1/4 (to the left of q₁) and 3/4 (to the right of q₁) . Similarly, the third quartile, q₃ , is the point located along the horizontal axis of the probability model which divides the probability mass into 3/4 (to the left of q₃) and 1/4 (to the right of q₃). Their difference,   iqr = q₃ - q₁, is called the interquartile range  of the probability model. The interquartile range is a parameter. The interquartile range is a scale or noise parameter. Locate the quartiles on your probability model above.

A second scale parameter is the population standard deviation , $\sigma$  . Recall that we gave a formula for it in the discrete case. We would have to use Calculus for the continuous case. But we can discuss it. The sample standard deviation, s , is an estimate of $\sigma$. Calculate your estimate.

Answers for the continuous probability model

Parameter	Parameter Value	Estimate Based on Sample
median	$\theta =$ 10	9.19
mean	$\mu =$ 10	11.14
First quartile	q1 = 5.84	6.31
Third quartile	q3 = 14.16	15.17
Interquartile range	q3 - q1 = 8.32	8.86
Standard deviation	$\sigma =$ 6	7.37