Expected value and variance binomial distribution June 24, 2004

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Expected value and variancebinomial
distributionJune 24, 2004
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Recall expected value
Discrete case
Continuous case
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Expected Value

• Expected value is an extremely useful concept for
  good decision-making!

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Example the lottery

• The Lottery (also known as a tax on people who
  are bad at math)
• A certain lottery works by picking 6 numbers from
  1 to 49. It costs 1.00 to play the lottery, and
  if you win, you win 2 million after taxes.
• If you play the lottery once, what are your
  expected winnings or losses?

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Lottery
Calculate the probability of winning in 1 try
The probability function (note, sums to 1.0)
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Expected Value
The probability function
Expected Value
E(X) P(win)2,000,000 P(lose)-1.00
2.0 x 106 7.2 x 10-8 .999999928 (-1) .144 -
.999999928 -.86  
Negative expected value is never good! You
shouldnt play if you expect to lose money!
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Expected Value
If you play the lottery every week for 10 years,
what are your expected winnings or losses?
  520 x (-.86) -447.20
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Empirical Mean(each person, cell, etc. counts
once)
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Variance/standard deviation

• Probability distributions not only have central
  tendency (means), but also have ranges (described
  by variance or standard deviation).
• Var(x) E(x-?)2
• The expected (or average) squared distance (or
  deviation) from the mean
• We square because squaring has better
  properties than absolute value. Take square root
  to get back linear average distance from the mean
  (standard deviation).

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Empirical Variance
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Binomial distribution

• Introduction
• Take the example of 5 coin tosses. Whats the
  probability that you flip exactly 3 heads in 5
  coin tosses?

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Binomial distribution

• Solution
• One way to get exactly 3 heads HHHTT
• Whats the probability of this exact arrangement?
• P(heads)xP(heads) xP(heads)xP(tails)xP(tails)
  (1/2)3 x (1/2)2
• Another way to get exactly 3 heads THHHT
• Probability of this exact outcome (1/2)1 x
  (1/2)3 x (1/2)1 (1/2)3 x (1/2)2

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Binomial distribution

• In fact, (1/2)3 x (1/2)2 is the probability of
  each unique outcome that has exactly 3 heads and
  2 tails.
• So, the overall probability of 3 heads and 2
  tails is
• (1/2)3 x (1/2)2 (1/2)3 x (1/2)2 (1/2)3 x
  (1/2)2 .. for as many unique arrangements as
  there arebut how many are there??

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The probability of each unique outcome (note
they are all equal)
ways to arrange 3 heads in 5 trials

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Binomial distribution functionX the number of
heads tossed in 5 coin tosses
p(x)

x
0
3
4
5
1
2
number of heads
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Binomial distribution, generally
Note the general pattern emerging ? if you have
only two possible outcomes (call them 1/0 or
yes/no or success/failure) in n independent
trials, then the probability of exactly r
successes
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Binomial distribution definitions
Binomial Suppose that n independent experiments,
or trials, are performed, where n is a fixed
number, and that each experiment results in a
success with probability p and a failure with
probability 1-p. The total number of successes,
X, is a binomial random variable with parameters
n and p We write X Bin (n, p) reads X
is distributed binomially with parameters n and
p   And the probability that Xr (i.e., that
there are exactly r successes) is   P(Xr)
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Binomial distribution
RECALL All probability distributions are
characterized by an expected value and a
variance   If X follows a binomial distribution
with parameters n and p X Bin (n, p)
Then   The expected value of a binomial
np The variance of a binomial np(1-p) The
standard deviation of a binomial
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Binomial distribution example

• If I toss a coin 20 times, whats the probability
  of getting exactly 10 heads?

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Binomial distribution example

• If I toss a coin 20 times, whats the probability
  of getting of getting 2 or less heads?

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In-Class Exercise
Suppose that exactly 55.1 of potential voters
who currently favor Kerry (a priori knowledge
that only we have!). NBC news conducts a poll
which consists of randomly calling 1000 eligible
voters and asking their voting preference,

• If the NBC researcher samples 1000 random voters,
  whats the probability that exactly 551 of them
  say that they favor Kerry?
• If the NBC researcher samples 1000 random voters,
  how many do you expect to say they favor Kerry
  (if someone is going to pay you a million dollars
  if you guess this right, whats your best guess?)
• Calculate the variance and standard deviation of
  the number of sampled voters (out of 1000) who
  vote yes on the recall.
• If the NBC researcher finds that 400 out of 1000
  of his random sample reported that they would
  voted yes for Kerry, what might you think about
  his sampling methods? (defend your opinion with
  numbers!)
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•  
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•  

 
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In-Class Exercise

• If the NBC researcher samples 1000 random voters,
  whats the probability that exactly 551 of them
  say that they favor Kerry?
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•  
•  

A very small number!
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In-Class Exercise
b. If the NBC researcher samples 1000 random
voters, how many do you expect to say they favor
Kerry (if someone is going to pay you a million
dollars if you guess this right, whats your best
guess?)      
Your best guess is 551. (1000x.551)
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In-Class Exercise
c. Calculate the variance and standard
deviation of the number of sampled voters (out of
1000) who would vote yes for Kerry.      
Variancenp(1-p)1000(.551)(.449)247.4 Standard
deviation square root (247.4)15.7
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In-Class Exercise
d. If the NBC researcher finds that 400 out of
1000 of his random sample reported that they
would vote yes for Kerry, what might you think
about his sampling methods? (defend your opinion
with numbers!)
EXPECTED DEVIATION 15.7 unlikely to see
deviation of 151 (which is so much greater than
the expected deviation) from the expected value
of 551
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Reading for this week

• Walker 1.1-1.2, pages 1-9

 
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Reading for next week

• Walker 1.3-1.6 (p. 10-22), Chapters 2 and 3 (p.
  23-54)