| Hand in: Sec. 4.4
A) Read "Prestidigitator of Digits", on reserve + outside my door (Due Monday): IPS presents the "relative frequency" foundation for probability. The same probability rules apply to "personal probabilities," (IPS p. 299) where an individual assigns fractions between 0 and 1 to the outcomes, giving some sort of "likelihood" that the outcome will happen. (they still have to sum to 1, P(not A) = 1-P(A), and P(A or B) = P(A) + P(B) if A and B have no outcomes in common.) Diaconis is again at Stanford, after a couple of years at Cornell. --What is the capitalized name given in the article to this kind of personal probability approach? --How does Diaconis feel when he sits in front of his paper? Sec. 4.4, p. 306 ff
DO 4.66 Benford's
law
(This is properly about probability as long-term average; put
in the
Law of Large Numbers section for some reason...)
|
Read, discuss
Play with Applet, called Expected value (Mean of a Random Variable) not Law of Large Numbers as book p.296 says. 4.71, 72 independent or not? Postpone 4.67
|
Optional
|
Law of large numbers: The average (mean) of the
values
of X observed in many (independent) trials approaches
µx,
as n gets larger . Parallel to probability as long-term
proportion.
Applet, called Expected value (Mean
of
a Random Variable) not Law of Large Numbers as book
p.296
says. Each graphed number on the zigzag line is the
average of all the "rolls" up to that point. Choose "Show roll totals"
to see blue dot giving that roll's result: below the zigzag line pulls
line (average) down, above pulls it up.
"Law of small numbers" = fallacy; run of good must be "made
up for" by run of bad, or vice versa. Runs are "swamped" by
mediocre
behavior, not compensated for.
µx: 1) balance point, 2) long run average of many independent observations.
Algebra of Means and Variances
Sometimes we change scale (Fo to Co),
sometimes we measure two things in the same experiment (height/weight,
red die/black die).
If X and Y are random variables, and we do linear
things to them, we can (often) find the mean and variance of the
result,
just
from knowing the original mean and variance (without having to find
the whole distribution of the result.) I'll call these rules
the Algebra of Means and Variances. (These rules hold
for discrete and continuous models.)
Means: p. 298 Variances:
p. 300
Rules 1 are for a linear transformation a+bX of one R.V. (cf.
p. 53-55 for data),
Rules 2 (&3) for a linear combination X + Y
Linear transformation W = a+bX: Just
changing the "ruler" at the base of the histogram or density
Mean: µw =
µ
a+bX =a +bµx
Variance: sigmaw2
= sigmaa+bX 2 = b2sigmax2
S.d. :
sigmaw = sigmaa+bX =
|b|sigmax
Example: X = # of heads on 2
flips
of the coin. µx= 1
sigmax2 = .5 sigmax
= .71
x|
2 | 1 | 0 |
P(X=x)
| .25| .50
| .25 |
Yolanda wins $2 for every head, and pays $1.50 to play.
Y = 2X-1.5. µY=
2·1-1.5
= .5 sigmaY2 = 22·.5=
2 sigmaY = |2|·.71 =
1.42
Will runs the game; pays $2 for every head and gets $1.50 from the
customer to play..
W = -2X +1.5. µw= -2·1+1.5
= -.5 sigmaw2 =(- 2)2·.5=
2 sigmaw = |-2|·.71 =
1.42
w|-2.5| -.5 | +1.5|
y| 2.5| .5 | -1.5|
x|
2 | 1 | 0 |
P(X=x)
| .25| .50
| .25 |
Start here Friday:
Linear combination:
If we have two random variables X and Y we will say
they
are "independent random variables" if every event involving
just X is independent of every event involving just Y.
(Usually we know this from the setup of the probability
model.
Roll 2 dice, outcome on red is independent of outcome on black.
Pick
a person, weight is not independent of height.)
W =X+Y: Scenario 1: Xavier flips a
coin
twice. Yancy flips a coin twice. Wendy counts how many
heads
total.
w| 0 | 1 | 2 |
3 | 4 | (p.
280-1: X is first 2 flips, Y second 2)
P(W=w)
| 1/16| 4/16 | 6/16 |
4/16 | 1/16 |
µw = 2 by symmetry.
= µx + µY
sigmaw2 =(0-2 )2 (1/16)+(1-2
)2 (4/16) +(2-2)2(6/16) + (3-2)2(4/16)
+ (4-2)2(1/16) = (4 + 1·4 + 0 + 1·4 +
4)/16
= 16/16 = 1 = sigmax2 + sigmaY2
sigmaw = 1 = sqrt(sigmax2
+ sigmaY2 ) Pythagorean
theorem
again...
W =X+Y: Scenario 2: Wendy flips a coin
twice.
Xavier pays her $1 for every head, Yancy pays her another $1 for every
head.
( Note W has same dist as 2X)
2|
4
1| 2
w| 0 | 2 | 4 |
0|_0________
P(W=w)
| .25 | .50
| .25 |
0 1 2
µw = 2 by symmetry.
= µx + µY
sigmaw2 = same as sigma2X2
= (2)2 .5 = 2, more than Scenario 1.
sigmaw = 1.414
Rule 2: µX + Y =
µx + µY whether
or
not X and Y are independent.
sigmaX+Y2 = sigmax2 +
sigmaY2
IF X
and
Y are independent.
Rule 3: sigmaX+Y2
= sigmax2 + sigmaY2 + 2·rho·
sigmax · sigmaY in general
Rho: theoretical analog of sample
correlation
coefficient. In Scenario 2, Rho = 1 (perfectly correlated).
Check: sigmax2 + sigmaY2
+ 2·rho· sigmax · sigmaY =
.5
+.5 + 2·1· .71·.71 = 1+
2·.5
= 2
General linear combinations:
X and Y are independent, µx =
5,
µY = 4, sigmax = 3, sigmaY
=2
W = 2X - 3Y + 4. Find mean and s.d.
Mean: 2·5 - 3·4 + 4 =
10-12+4 = 2
S.d.: First find variance. sigmax2
= 9 , sigmaY2 = 4
sigmaw2 = 22·sigmax2
+ (-3)2·sigmaY2 =
4·9
+9·4 = 72
sigmaw = sqrt(72) = 6·1.414 = 8.484
Note sigmaX-Y2=
sigmax2 +
sigmaY2
| Sievers home | Math251-Fall05/Dayps23.htm | 10:50 am | 10/19/05 |