Comment on Activity 1

We show the graph of the Cauchy density function in Figure 1 in the text. This graph suggests that data values selected at random would be tightly clustered around `0`. For example, for two different intervals of length `1` we have

`int_(-0.5)^(0.5) f text[(] t text[)] dt = 0.295`   and   `int_5^6 f text[(] t text[)] dt = 0.010`.

Thus it is almost `30` times more likely to find a data value in the interval `[-0.5,0.5]` as to find one in `[5,6]`.

We showed in Example 5 in the preceding section that `int_(- oo)^(oo) 1/(1+t^2) dt = pi`, so `int_(- oo)^(oo) f text[(] t text[)] dt = 1`. For any density function `f`, the integral over the entire domain represents the probability that a random data value lies somewhere in the entire set of data values, and this probability must be `1`.

We find the distribution function `F` by integrating `f` from the left end of the domain up to an arbitrary `t`:

`F text[(] t text[)]` `= 1/(pi) int_(- oo)^t 1/(1+s^2) ds= lim_(T rarr - oo) 1/(pi) int_T^t 1/(1+ s^2) ds`
  `= 1/(pi)  lim_(T rarr - oo) (tan^(-1) t - tan^(-1) T)`
  `= 1/(pi) [tan^(-1) t - (- (pi)/2)]=1/2 + 1/(pi)  tan^(-1) t`.

We see from this formula that

`lim_(t rarr - oo) F text[(] t text[)] = 1/2 + 1/(pi)(-(pi)/2)=0`   and   `lim_(t rarr oo) F text[(] t text[)] = 1/2 + 1/(pi)((pi)/2)=1`.

Also, `F` is an increasing function because its derivative `f` is always positive.