Parameter and Parametric Space – Example

Let me explain the concept with the help of an example.  The biophysical process behind catching a disease say Breast Cancer has a deterministic and probabilistic aspect to it. The deterministic aspect can be explained (determined) and controlled. Having a healthy and balanced diet, doing regular exercises etc are some deterministic examples of controlling breast cancer. But several other unexplained factors including the psychology of an individual, genetic makeup etc contribute to the probabilistic or random facet of this disease. This random aspect has to be explained by probability distribution function. So if the chance of getting cancer has to be quantified with a numerical value and described by an equation; then there will be a deterministic part and a probabilistic part. The parameters and the parametric space of this biophysical phenomenon have to be identified for an exact estimation of this probability. The probability distribution will be a multivariate distribution as multiple random variables contribute to this process.

Normal distribution can explain various indicators of health condition like blood pressure, cholesterol, sugar etc. Normal distribution tells that mean or average value has the highest probability and as we go away from the mean on either side the probability decreases. For this normal distribution the figure below gives Normal probability density function for different values of the parameter µ and σ². Here each curve could depict say blood pressure for each age group. The normal curve on the left most side could represent the probability density function of blood pressure pattern of age 20- 25. The normal curve in the middle could represent blood pressure pattern of age 40 – 45.  The normal curve of the right could represent the blood pressure pattern of age 60 – 65. The average blood pressure is lowest for the age 20-25. The variance of blood pressure around the mean is lowest for the age 20-25.