From Algorithms to Z-Scores - matloff - University of California, Davis

5.5. FAMOUS PARAMETRIC FAMILIES OF CONTINUOUS DISTRIBUTIONS 101
V ar(X) = E(X 2 ) − (EX) 2
=
4
1
t 2 2t/15 dt − 2.8 2
(from (3.30) (5.28)
(from (5.25)) (5.29)
= 5.7 (5.30)
Suppose L is the lifetime of a light bulb (say in years), with the density that X has above. Let’s
find some quantities in that context:
Proportion of bulbs with lifetime less than the mean lifetime:
Mean of 1/L:
P (L < 2.8) =
2.8
1
E(1/L) =
4
1
2t/15 dt = (2.8 2 − 1)/15 (5.31)
1
2
· 2t/15 dt =
t 5
(5.32)
In testing many bulbs, mean number of bulbs that it takes to find two that have
lifetimes longer than 2.5:
Use (3.114) with r = 2 and p = 0.65.
5.5 Famous Parametric Families of Continuous Distributions
5.5.1 The Uniform Distributions
5.5.1.1 Density and Properties
In our dart example, we can imagine throwing the dart at the interval (q,r) (so this will be a
two-parameter family). Then to be a uniform distribution, i.e. with all the points being “equally
likely,” the density must be constant in that interval. But it also must integrate to 1 [see (5.19).
So, that constant must be 1 divided by the length of the interval:
for t in (q,r), 0 elsewhere.
fD(t) = 1
r − q
(5.33)