Homework #2 ECES 490
1. On planet Vulcan, there are an equal number of adults (men and women) and children.
Men’s heights are uniformly distributed between 18” and 36”. Women’s heights are
uniformly distributed between 54” and 72”. Among adults, there are an equal number of
men and women. Children have heights uniformly distributed from 6”-12” (children
instantly grow to their adult height when they turn into adults at age 75). On Vulcan, even
if you know a person’s name, you cannot tell whether that person is a man or a woman
unless you have more information.
A Vulcan creates a message containing the following information about him or her:
Message to be communicated:
Name:
Sex:
Age:
Height:
When this message is received, at a distant location, it is not completely readable (due to
impairments that are introduced in the communication process).
You are trying to build a machine that uses the received message to estimate the height of
the Vulcan who created the transmitted message, using minimum mean squared error as a
criterion for the goodness of the estimate.
What are the minimum-mean-square-error estimators* for the following cases?
A.
B.
C.
D.
The received message is completely readable
Only the field with a name of the Vulcan is readable
Only the field with the age of the Vulcan is readable
Only the field with the sex of the Vulcan is readable
*The estimator is the algorithm that specifies how to translate the received information
into an estimate of the Vulcan’s height. For example, in case C. the estimator has the
form
If the number in the age field is <75, then the minimum mean square error estimate is …
If the number in the age field is >75, then the minimum mean square error estimate is …
If the number in the age field is 75, then the minimum mean square error estimate is …
Solutions
A. If the received message is completely readable then we know the exact height of the
subject. Thus, the minimum mean square error estimate of the height is the number in
the height field.
B. If only the name of the Vulcan is readable then we don’t know anything about the
subject regarding its height. In that case there is a 50% chance that it is a child and a
50% chance that it is an adult. The average height of a child is 9 inches. The average
height of a male adult is 27 inches. The average height of a female adult is 63 inches.
The average height of an adult is (0.5 x 27) + (0.5 x 63) = 45 inches. The average
height of a Vulcan is (0.5 x 9) + (0.5 x 45) = 27 inches. Thus the minimum mean
square error estimate of the height in this case is 27 inches.
C. If the number in the age field is less than 75 then we know that the subject is a child
and, consequently, it has a height between 6 and 12 inches, with a uniform probability
distribution. The mean (average) of this probability distribution is 9 inches. Thus the
minimum mean square error estimate of the height of the subject if the number in the
age field is 75 is 9 inches. If the number in the age field is greater or equal to 75
then we know the subject is an adult. Thus there is 50% chance that the subject is a
male adult and a 50% chance that the subject is a female adult. The average height of
a male adult is 27 inches. The average height of a female adult is 63 inches. The
average height of a Vulcan adult is (0.5 x 27) + (0.5 x 63) = 45 inches. Therefore, the
minimum mean square error estimate of the height of the subject if the number in the
age field is 75 is 45 inches.
D. If we only know that the subject is a Vulcan female, then we know that there is a 50%
chance that it is a child (girl) and a 50% chance that it is a female adult. The average
height of a child is 9”. The average height of a female adult is 63”. The average
height of a Vulcan female is (0.5 x 9) + (0.5 x 63) = 36 inches. Thus the minimum
mean square error estimate of the height of a Vulcan female is 36 inches. Similarly, if
we only know that the subject is a Vulcan male, then we know that there is a 50%
chance that it is a child (boy) and a 50% chance that it is a male adult. The average
height of a child is 9”. The average height of a male adult is 27”. The average height
of a Vulcan male is (0.5 x 9) + (0.5 x 27) = 18 inches. Thus the minimum mean
square error estimate of the height of a Vulcan male is 18 inches.