Transcript The Normal Approximation for Data

The Law of Averages
What does the law of average say?
• We know that, from the definition of probability, in the long run the
frequency of some event will be around the probability of the event.
• For example, a coin lands heads with probability 50%. So after many
tosses, the number of heads should approximately equal the number
of tails.
• Is the law of average all about this?
Question
• Suppose we are tossing a coin. If we get a lot of heads, then tails start
coming up. Or if we get too many tails, the chance for heads goes up.
In the long run, the number of heads and the number of tails even
out.
• True or false? Why?
• Answer: False.
Explanation
• With a fair coin the chance for heads stays at 50%, no matter what
happens. We learned this from independence.
• So whether there are two heads in a row or twenty, the chance of
getting a head next time is still 50%.
• Moreover, since the probability keep fixed, if we get a lot heads at the
beginning, the number of heads and the number of tails may not
even out at the end.
• Instead, the chance error will go up in absolute terms. Here chance
error means the amount off the expected value. For the tossing coin
example, number of heads = half the number of tosses + chance
error.
Explanation
• This will not contradict to the probability. Because in the long run the
chance error only gets bigger in absolute terms, however, compared
to the number of tosses, it gets smaller.
• For example, in an experiment a coin was tossed for 30, 500, 5000,
10000 times, the chance error was 2, 5, 33, 67. This shows the chance
error gets bigger. But the percentage gets smaller 6.67%, 1.00%,
0.66%, 0.67%.
• A natural question will be: how big the chance error is likely to be?
• For example, with 100 tosses, the chance error is likely to be around 5
in size. With 10,000 tosses, the chance error is likely to be around 50
in size. Multiplying the number of tosses by 100 only multiplies the
likely size of the chance error by √100 = 10. (We will discuss this later.)
The table for the experiment
The graphs for the experiment
Summary of the law of averages
• As the number of tosses goes up, the difference between the number
of heads and the expected number (half the number of tosses) gets
bigger. That is the chance error gets bigger.
• But the difference between the percentage of heads and the
probability (50%) gets smaller. That is the ratio between the chance
error and the total number of tosses gets smaller.
• The likely size of the chance error will get bigger, but the rate will not
be as fast as that of the total number of tosses.
Chance processes
• In the previous example, we met with the problem of chance
variability: when a coin is tossed a large number of times, the actual
number of heads is likely differ from the expected number.
• Until now, we still don’t know how to calculate the likely size of the
difference (chance error).
• These problems are about chance processes: chance comes in with
each toss of the coin. If we repeat the experiment, the tosses turn out
differently, and so does the number of heads.
Chance processes
• In order to study all kinds of chance processes (e.g. tossing a coin, playing
the roulette, drawing a random sample from a large population), we need
to generalize a standard model to analyze how the numbers influenced by
chance:
• The box model----drawing numbers at random from a box. (It is easier for
us to analyze mathematically.)
• The idea is that:
• Find an analogy between the chance process and the box model.
• Connect the variability with the chance variability in the sum of the
numbers drawn from the box.
• We will see how to use this idea from the examples later.
The sum of draws
• Let us look at our standard model process:
• There is a box of tickets.
• Each ticket has a number written on it.
• Some tickets are drawn at random from the box, and the numbers on
these tickets are added up.
Example
• Suppose we have a box with tickets: 1, 2, 3, 4, 5, 6.
• We draw twice at random with replacement from this box.
• We then add up the two numbers.
• For example, we may draw 3 at first, then 5 the second. So the sum is
8. Or the first draw may be 3, and the second may be 3 again. So the
sum is 6.
• There are many other possibilities. The sum is subject to chance
variability.
• Notice that this process is the box model for rolling a pair of dice,
then add up the two numbers.
Another example
• Now we make 25 draws from the same box: 1, 2, 3, 4, 5, 6. Then study
the sum of the numbers drawn from the box.
• We programmed the computer to make the draws:
• 32462 35442
36412
41556
22255
• The sum is 88.
• If we had the computer repeat the whole process ten times, the sum
would have been different:
• 88 84 80 90 83 78 95 94 80 89
• The chance variability is easy to see. The values range from 78 to 95.
Another example
• In principle, the sum could have been as small as 25 x 1 = 25, or as
large as 25 x 6 = 150.
• But in fact, the ten observed values are all between 75 and 100.
• Would this keep up with more repetitions?
• What is the chance that the sum turns out to be between 75 and
100?
• We will study this kind of problem later in the course.
• This box model tells us how to use the simplified model to analyze the
process we want to know about.
Making a box model
• The sum of the draws from the box turns out to be the key ingredient
for many statistical procedures, so keep your eye on the sum.
• Before we make a box model, try to ask the following question first:
• What numbers go into the box?
• How many of each kind?
• How many draws?
• The purpose of a box model is to analyze chance variability, which can
be seen in its starkest form at any gambling casino. So let us focus on
the box models for roulette.
The Nevada roulette
• A Nevada roulette wheel has 38 pockets:
• One is numbered 0, another is numbered 00, and the rest are
numbered from 1 through 36.
• The croupier spins the wheel, and throws a ball onto the wheel.
• The ball is equally likely to land in any one of the 38 pockets.
• Before it lands, bets can be placed on the table.
A Nevada roulette table
• Let us focus on the red or black first.
• Red or black: except for 0 and 00, which
are colored green, the numbers on the
roulette wheel alternate red and black.
• If you bet a dollar on red, say, and a red
number comes up, you get the dollar
back together with another dollar in
winnings. If a black or green number
comes up, the croupier smiles and rakes
in your dollar.
• The case that you bet a dollar on black is
similar.
Convert to a box model
• If you bet a dollar on red, and the croupier spins the wheel. It may seem
hard to figure your chances. But a box model will help.
• The numbers go into the box:
• You will either win a dollar or lose a dollar. So the tickets must be either +1
or -1.
• How many of each kind:
• We know 0 and 00 are green, and numbers from 1 through 36 alternate red
and black. So you win if one of the 18 red numbers comes up, and lose
otherwise. Therefore, your winning chance is only 18 in 38, and the chance
of losing is 20 in 38.
• There are 18 “+1” tickets, and 20 “-1” tickets in the box.
Convert to a box model
• As far as the chances are concerned, betting a dollar on red is just like
drawing a ticket at random from the box.
• The advantage of the box model is that all the irrelevant details----the
wheel, the table, and the croupier’s smile----have been stripped away.
• The cruel reality is that you only have 18 tickets and they have 20.
The net gain
• Suppose you play roulette ten times, betting a dollar on red each
time.
• You will end up ahead or behind by some amount. This amount is
called your net gain.
• The net gain is positive if you come out ahead, negative if you come
out behind.
Box model for net gain
• On each play, you win or lose some amount. This is modeled by
drawing a ticket from a box.
• In ten play, it is just like ten draws from the box, made at random with
replacement.
• So the net gain----the total amount won or lost----is just the sum of
these ten win-lose numbers.
• So the box model will be the sum of ten draws made at random with
replacement from the box with 18 “+1” tickets and 20 “-1” tickets.
Example
• Suppose, for instance, that the ten plays came out this way:
•R R R B G R R B B R
• (R for red, B for black, and G for green.)
• We have the following table for the net gain:
When you get a red, the win-lose number is +1, and the net gain goes up by 1. When you get a
black or a green, the win-lose number is -1, and the net gain goes down by 1. The net gain is
just the sum of the win-lose numbers, which is just like the sum of draws. This example has a
happy ending: you come out ahead $2. We will study later about the chance if you keep on
playing.
Another example
• When you bet a dollar on a single number at Nevada roulette, and
that number comes up, you get the $1 back together with winnings of
$35.
• If any other number comes up, you lose the dollar.
• Gamblers say that a single number pays 35 to 1.
• Suppose you play roulette 100 times, betting a dollar on the number
17 each time. What is the box model?
Solution
• The first question is what numbers go into the box:
• In one play, you will win $35 dollars only when the ball drops into the
pocket 17, otherwise you lose $1.
• So the tickets will be “+35” and “-1”.
• Comment: The tickets in the box show the various amounts that can
be won or lost on a single play.
Solution
• The second question is how many tickets of each kind:
• In one play, you win only when then number 17 comes up. It is only 1
chance in 38 of winning. For the other 37 chances in 38, you lose.
• So there is 1 ticket for “+35” and 37 tickets for “-1”.
• Comment: The chance of drawing any particular number from the box
must equal the chance of winning that amount on a single play.
(“Winning” a negative amount is the mathematical equivalent of
what most people call losing.)
Solution
• The third question is how many draws:
• Since you are playing 100 times, the number of draws has to be the
same----100.
• Tickets must be replaced after each draw, so as not to change the
odds. That is we draw ticket with replacement.
• Comment: The number of draws equals the number of plays.
Solution
• So the net gain in 100 plays is like the box model:
• Sum of 100 draws made at random with replacement from the box
with 1 ticket for “+35” and 37 tickets for “-1”.
Summary
• There is chance error in the number of heads:
• Number of heads = expected value + chance error.
• The error is likely to be large in absolute terms, but small relative to
the number of tosses. This is the law of averages.
• In percentage terms, the percentage of heads is likely to be close to
50%, although it is not likely to be exactly equal to 50%.
• The law of averages does not work by changing the chances. For
example, after a run of heads in coin tossing, a head is still just as
likely as a tail.
Summary
• A complicated chance process for generating a number can often be
modeled by drawing from a box. The sum of the draws is a key
ingredient.
• The basic questions to ask when making a box model:
• What numbers go into the box?
• How many of each kind?
• How many draws?
Summary
• For gambling problems in which the same bet is made several times, a
box model can be set up as follows:
• The tickets in the box show the amounts that can be won(+) or lost(-)
on each play.
• The chance of drawing any particular value from the box equals the
chance of winning that amount on a single play.
• The number of draws equals the number of plays.
• The net gain is like the sum of the draws from the box.