Following are examples to "convince" you of rules 1 and 2 of the "algebra of means and variances." As you work them, try to look at things in your work that would make rule 1 or 2 work "in general."
A. Linear transformation: Suppose you have an urn (bowl)
with 4 balls in it, labeled
1, 1, 2, 4. Choose a ball at random. X = number on ball. (My
fairy godmother Xena lets me draw a ball each day, and gives me that
many dollars)
Then P(1) = 1/2, P(2) = 1/4, P(3) = 1/4. We find
that the mean of X = 2, and the variance = 1.5. (Examples 4.20,
21,
24 show how.)
Let W = 3X (Grandfather Walter banishes the fairy
godmother, and gives me 3 times whatever the ball says, each day)
Let Y = 5 + W = 5 + 3X. Aunt Yetta gives me $5 a day in
addition to Walter's gift.
Give the probability distributions of W and Y. (in a
table.)
Graph the probability histogram of X, and draw new number
axes under it showing the distributions of W and Y.
Find the means of W and Y. Find the variances and
standard
deviations of W and Y.
Check whether rule 1 for means (p. 298) holds. (a = 5,
b = 3 here)
Check whether rule 1 for variances (p. 302) holds.
Linear combination X+Y
B. Suppose you have an urn with 4 chips in it, black
on one side, red on the other. They have numbers on each side, as
follows (one of the chips with 10 on its black side has 1 on its red
side
, etc.)
X: Black: 10 0
4
10
Y: Red : 1
1
2 4
The experiment is to choose a chip at random. Let X= number on
black side, Y=number on red side, W = X+Y.
Write down the probability distributions for X, Y, and W.
Make up a story (perhaps involving Xena, Yetta and Walter)
about
this urn.
Find the means of X, Y, W. (You do not need to repeat
any work you
have done for part B; just refer to it.)
Find the variances of X, Y, W.
Check whether rule 2 for means (p.298) holds.
Check whether rule 2 for variances (p. 302) holds.
Note: X and Y are clearly not independent--if
you tell me you drew a chip with black side 4, I know the red side is 2.
C. Suppose you have one urn with 4 black balls,
labeled
0, 4, 10, 10, and another! urn with 4 red balls, labeled 1, 1, 2, 4 (yes,
this is the fairy godmother's urn, but Yetta has it now). You get
to draw one ball from
each urn. X= number on black ball, Y = number on red
ball.
Satisfy yourself that you have "already" found the mean and
variance
for X and Y.
W = X + Y Find the probability distribution for W. (This
is more complicated than B. There are 16 possible pairings of
balls,
equally likely, but some have the same numbers on them. Express the
probabilities
as sixteenths. This grid may be helpful. Or a tree, black
ball
followed by red ball.)
y\\x 0 4 10 10
4 * * * *
2 * * * *
1 * * * *
1 * * *
*
(Hint: S = {1,2,4,5,6,8,11,12,14}) *If you
can't
get it, go here, get the
answers
and go on.
Find the mean and variance for W.
Check whether rule 2 for means (p. 298) holds.
Check whether rule 2 for variances (p. 302) holds.
Why is it clear from the probability mechanism that rule 2 should work
here?
Rule 2 and Rule 3 for variances differ in the "tag" involving rho,
the correlation coefficient, in the dependent situation. Use your
results from parts B and C and algebra to calculate the value of rho
in part B. (There is a formula, parallel to the one
for r, the sample correlation coefficient. Math 300)
Make a "scatterplot" on x/y axes of the four pairs of
values in part B. If this were a data set, would rho (r) have the
right sign?
Linear combination X - Y
D. Some people are surprised by the second part of rule 2 for
variances (p.302)
(variance of difference = sum of variances.)
Using the setup of problem C, let Z = X - Y (Zelda, the semi-good
fairy, offers me this game--I draw 2 balls each day, one from each
urn.
I compute X - Y. If the number is positive, she pays me that
much;
if it is negative, I pay her...)
Find the probability distribution for Z. (Hint: S = {
-4, -2, -1, 0, 2, 3, 6, 8, 9}) *If you can't get it, go
to
here, get the answers and go
on.
Find the mean and variance for Z, check that the rules hold.
Now: Think of Z = X + (-1)Y. I.e. "b" from the a+bX
rule
is -1.
Check that combining rules 1: µ a+bX
=a +bµx, and first
part
of 2: µ (X+Y) =µX
+ µY for means
(p. 298) give the correct
answer for the mean of Z.
Check that the second part of rule 2 for variances (p.302) holds,
combining rule1 and the first part of rule 2 for variances.
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