Supplementary Exercise 4.56 of IPS7e
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We denote by Y a random variable with a uniform distribution on the
interval (0,2).


(a) The height of the density curve is 0.5 (in order to obtain an area
under the curve of 0.5*2=1). The graph should have a horizontal line at
the value of 0.5 in the interval (0,2), and be zero elsewhere.

Minitab can produce a Probability Distribution Plot (in the Graph menu)
for a uniform distribution:

MTB > DPlot;
SUBC>   Distribution;
SUBC>     Uniform 0.0 2.0


(b) The area under the curve for the interval (0,1) is 0.5*1=0.5. We could write
this result as
    P(Y<1) = 0.5.


(c) Again we compute the area under the curve.
    P(0.5<Y<1.3) = 0.5*(1.3-0.5) = 0.4.


(d) P(Y>=0.8) = 0.5*(2-0.8) = 0.6.

Minitab can also calculate this probability in a Probability
Distribution Plot (choose View Probability, and use the Shaded Area tab
to define the area of interest. 

MTB > DPlot;
SUBC>   Scale 1;
SUBC>     LDisplay 1 0 0 0;
SUBC>   Distribution;
SUBC>     Uniform 0.0 2.0;
SUBC>   Shade 1;
SUBC>     ShType 2;
SUBC>     ShValue 0.8.
Distribution Plot 

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Note that it does not matter for a continuous distribution (i.e., with a
density function) whether the boundaries of the intervals are included
or not. We could therefore also have written P(Y<=1), P(0.5<=Y<=1.3) and
P(Y>0.8), and the results would have been exactly the same.