[The Dart Model] Suppose you are
throwing a dart at a region in the plane and measuring a random variable
X determined by the position of the dart. X has values only between 0 and
10. Unfortunately you are not given the distribution function of X. [Recall
that for any real number A between 0 and 10 the distribution function F,
would give you F(A), the probability that the value of X is less than or
equal to A.]
Instead you have been given a graph, an algebraic expression and a
table for the probability density function of X. [Recall that for
any real number A between 0 and 10 the probability density function
of X, f, gives you f(A), the derivative of F at A,
which is approximately the ratio of the probability of the random variable
X being in an interval [A, A+h] to the length of that interval,
h.]
Based on this information do you think you could estimate the probability
that the value of X being less 5? Recognizing that F(0) = 0 and F(10) =
1, could you estimate F(A) for any A between 0 and 10? As a practical matter,
you might break the interval into10 pieces of equal length and then estimate
the probability of X being in each of those interval from the density of
X at some point in that interval. Then for A, just add up the probability
estimates for X being in an interval that has values less than or equal
to A. 
A 
f (A) 
0.5 
.05 
1.5 
.13 
2.5 
.11 
3.5 
.15 
4.5 
.19 
5.5 
.17 
6.5 
.09 
7.5 
.07 
8.5 
.03 
9.5 
.01 

Graph of data for f(A)
The Probability Density Function for X

Solution: As suggested, consider
the interval [0,10] broken into smaller intervals each of length 1, i.e.,
[0,1], [1,2], [2,3], ... , [8,9], and [9,10]. Since each of these intervals
has length 1, we can use f (A) as an estimate for the probability
of the dart falling in the inteval that contains A. So, for example we
estimate the probability that X lies in the interval [0,1] with f
(.5) = .05, while we estimate the probability that X lies in the interval
[1,2] with f (1.5) = .13. Continuing in this fashion we make estimates
of the probability of X being in each of the ten intervals.
Now use these to estimate the probability that X is less than 5 by accumulating
the probabilities of the intervals that would have X less than 5. Thus
we estimate
F(5) = Prob( X<5) = 0.05 + 0.13 + 0.11 + 0.15 + 0.19 =
0.64 .
The distribution function might be estimated by assuming the density is
uniform for each of the ten intervals. So, for example, if 0<A<1
then F(A) = 0.05 A, while if 1<A<2 then F(A) = 0.05 + 0.13(A1).
Following this pattern if 9<A<10 then F(A) = 0.99 + .01(A9).
One way to visualize this example is to use the density function value
for the height of a rectangle with a unit base. Then the region in the
plane above the interval [0,10} has a total area of one square unit. Assuming
the distribution is uniform for each interval allows us to see the probability
that X < A as the area of the region above the X axis and bounded by
the vertical lines, X = 0 and X = A. 
Estimated probabilities that X is in
an interval based on the given probability densities
interval 
Estimated probability 
[0,1]

.05

[1,2]

.13

[2,3]

.11

[3,4]

.15

[4,5]

.19

[5,6]

.17

[6,7]

.09

[7,8]

.07

[8,9]

.03

[9,10]

.01


Estimated probability distribution function
based on the given probability densities
A in the interval 
F(A)

[0,1]

.05 A

[1,2]

.05 +.13 (A  1)

[2,3]

0.18 + .11 (A  2)

[3,4]

0.29 + .15 (A  3)

[4,5]

0.44 + .19 (A  4)

[5,6]

0.63 + .17 (A  5)

[6,7]

0.80 + .09 (A  6)

[7,8]

0.89 + .07 (A  7)

[8,9]

0.96 + .03 (A  8)

[9,10]

0.99 + .01 (A  9)

Graph of Estimated
Probability Distribution Function for
X
based on the given probability densities
