### 11. CONFIDENCE INTERVALS AND THEIR ARBITRARINESS

So far we have worked only in terms of relative probabilities and rms values to give an idea of the accuracy of the determination = *. One can also ask the question, What is the probability that lies between two certain values such as ' and "? This is called a confidence interval, Unfortunately such a probability depends on the arbitrary choice of what quantity is chosen for . To show this consider the area under the tail of ( ).  Figure 3. Shaded area is P( > '). (Sometimes called the confidence limit of '.)

If = ( ) had been chosen as the physical parameter instead, the same confidence interval is Thus, in general, the numerical value of a confidence interval depends on the choice of the physical parameter. This is also true to some extent in evaluating  . Only the maximum likelihood solution and the relative probabilities are unaffected by the choice of . For Gaussian distributions, confidence intervals can be evaluated by using tables of the probability integral. Tables of cumulative binomial distributions and cumulative Poisson distributions are also available. Appendix V contains a plot of the cumulative Gaussian distribution.