Page 102 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 85

It states that, in order to determine EfXg, it can be found by taking a weighted
average of the conditional expectation of X given Y ˆ y i ; each of these terms is
weighted by probability P Y ˆ y i ).

Example 4.8. Problem: the survival of a motorist stranded in a snowstorm
depends on which of the three directions the motorist chooses to walk. The first
road leads to safety after one hour of travel, the second leads to safety after
three hours of travel, but the third will circle back to the original spot after two
hours. Determine the average time to safety if the motorist is equally likely to
choose any one of the roads.
Answer: let Y ˆ 1, 2, and 3 be the events that the motorist chooses the first,
second and third road, respectively. Then P Y ˆ i) ˆ 1/3 for i ˆ 1, 2, 3. Let X
be the time to safety, in hours. We have:

3
X
EfXgˆ EfXjY ˆ igP Y ˆ i†
iˆ1
3
1 X
ˆ EfXjY ˆ ig:
3
iˆ1
Now,

EfXjY ˆ 1gˆ 1; 9
>
>
>
=
EfXjY ˆ 2gˆ 3; 4:16†
>
>
>
EfXjY ˆ 3gˆ 2 ‡ EfXg: ;
Hence
1
EfXgˆ 1 ‡ 3 ‡ 2 ‡ EfXg†;
3
or

EfXgˆ 3 hours:

Let us remark that the third relation in Equations (4.16) is obtained by noting
that, if the motorist chooses the third road, then it takes two hours to find that
he or she is back to the starting point and the problem is as before. Hence, the
motorist’s expected additional time to safety is just EfXg. The result is thus
2 ‡ EfXg. We further remark that problems of this type would require much
more work were other approaches to be used.

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