| Hand in: Wednesday (From Moore except as noted)
Prep for sec. 4.3: p. 241, 4.40 do a and b; Do b this way. Close your eyes and put your finger down somewhere on table B. Start reading the table where your fingertip lands. Find xbar for your sample. Now Repeat part b, to get a total of 3 values of xbar. (You can just keep reading the table where you left off, or you can put your finger in a different spot). Make a dotplot of your 3 values and bring the values to class to be compiled with everyone else's. Rest of 4.2:
Continuous sample spaces:
|
Read, to discuss
|
Optional
D. Law of Large numbers: Extra credit, hand in on separate sheet! Due Friday. LLN-game |
Chance behavior (a random phenomenon): Unpredictable in the short run, predictable regular pattern in the long run.
"Probability" of particular something happening:
proportion
of times it would happen in a very long series of independent
repetitions
of the phenomenon.
(independence:
outcome of one trial (repetition) must not influence the outcome of any
other.)
http://www.whfreeman.com/scc
What is probability? 1 toss at a time--settles down slowly.
Sec. 4.2 Probability Models
Random phenomenon,
Sample space S: set
of all possible outcomes (no overlap of descriptions)
Event: any outcome
or set of outcomes
Probability model:
S, and a way of assigning a probability to each event.
Probability rules: pp. 222-3, in
words, then in notation.
A an event in sample space S, P(A)
is "the probability that A occurs"
These rules are all true for
proportions
in long run (Probabilities), prop.of counts, proportions of areas.
1. 0 <
P(A) < 1
2. P(S) = 1
3. For any event A,
P(A
does not occur) = 1 - P(A)
4. A and B are
disjoint if they have no outcomes in common (can't happen simultaneously.)
If
A and B are disjoint, their probabilities add: P(A or B) = P(A)
+ P(B)
Finite sample spaces:
Assign a probability to each outcome (>0)
so they add to 1. (Sometimes equal values make sense.)
Prob. of an event is sum of
prob's of its outcomes.
Phenomenon: Flip coin twice.
S1 = {HH, HT, TH,
TT} S2
= {0, 1, 2} number of heads
S3 = {Y, N} both are heads?
Sample space | HH | HT | TH |
TT
|
Prob's
|
.25| .25| .25| .25| P(tail followed by head)=?
Sample space | 2 |
1 | 0 | P(at
least 1 tail)=? P(1 of each) = ?
Prob's
|
.25| .50 | .25| P(at
least 1 Head)= ? P(2Heads) = ?
Sample space | Y |
N |
Prob's
|
.25| .75 |
Often the sample space is naturally expressed in numbers, thus
Random Variable:
(X, Y, Z...) Variable whose value is a numerical outcome of a random
phenomenon.
Probability distribution of X tells
us what values X can take and how to assign probabilities to them.
If X has a finite number of
possible values (Discrete distributions), nothing new except notation.
P(X < 2) is "Prob.
that X is less than 2."
Flip coin twice. R.V. X
= number of heads:
Distribution given by table.
x| 2 | 1 | 0 |
P(X=x)
|
.25| .50 | .25|
P(X >
1) = ?
Words: Prob that #
heads is >
1
P(X = 2)
=
?
Prob that # heads is
2
Random variables with intervals of outcomes ("continuous")
Sec.
4.2 pp. 228-232
If the sample space is an interval of values (or the whole line),
the way we assign probabilities to events is with a density curve (cf.
Sec.
1.3, Day
8) (remember density curves were idealizations of
histograms--of repeating the "experiment" many many times)
P(a < X < b) = the
probability that X is between a and b is the area
under the density curve, between a and b.
We declare P (X = a) = 0 , so P(a < X < b)
= P(a < X < b)
Notation: Use capital letter for the random variable, the "label"
of the phenomenon. Use small letters for particular values it can
have. But this rule is often broken--Moore uses x-bar where many
would use X-bar.
B. Y = (number you get from) the sum of two spinners. ("Triangular")
a) The probability that the sum is a number less than .6 =
P( ?
) =.18
b) P(Y > 1.6) = ? P(Y < 1.6)
= ? P (Y < 1)
= ?
P( 1 < Y < 1.6) = ?
c) P(Y > x) = .08. Find x: ?
Our most important probability model is the NORMAL DISTRIBUTION
family. You use the same techniques as before, only we ask "probability
that one..." instead of "proportion of all..."
Law of Large Numbers (p.237, "LLN") Take observations at
random from a population with population mean µ.
Then as the number of observations n increases, the sample mean xbar
gets
closer and closer to µ. (Even
if the population is infinite!
Note--we don't say how big n needs to be for how close here.)
OR Let the sample size n get bigger. Then the xbars
will eventually get very close to the population mean µ.
OR As the sample size increases, the sample mean gets closer
to the population mean µ.
OR For a very large sample, the sample mean will (almost certainly)
be very close to the population mean.
Activstats p 15-3 activities.
Now: keep a fixed sample size n:
What probability distribution describes the random phenomenon of
finding xbar from a SRS?
That is, what is the distribution of the random variable Xbar,
when the experiment is to take a simple random sample of size n? We'll
call it the "sampling distribution of the sample mean" (Sec.
4.3)
(Spinning penny: sampling
dist. of proportion)
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