Transcript Lecture 18 Binomial distritution

Binomial random variables
Coin Toss Example
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If you toss a fair coin three time and let X= the
number of heads observed. Find the expected
value and variance of X.
There are different ways to solve this problem.
 From the three tosses, we have a total of 8
outcomes.
 {HHH, HHT, HTH, THH, HTT, THT, TTH,
TTT}
 Each of the above 8 outcomes has a
probability of 1/8.
Coin Toss Example
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One way of finding the mean is to count
the number of heads in each outcome and
take the average.
 {3,
2, 2, 2, 1, 1, 1, 0}
 The mean is therefore 12/8=1.5
 Then we can find the variance of the 8
numbers, which is:
 [(3-1.5)^2+3*(2-1.5)^2+3*(1-1.5)^2+(01.5)^2]/8=0.75
Coin Toss Example
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Another way is to find the pmf.
X
0
1
2
3
P(X)
1/8
3/8
3/8
1/8
E(X)=0*(1/8)+1*(3/8)+2*(3/8)+3*(1/8)=1.5
 Var(X)=(0-1.5)^2*(1/8)+(1-1.5)^2*(3/8)+(21.5)^2*(3/8)+(3-1.5)^2*(1/8)=0.75
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Coin Toss Example
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Yet another way.
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Here, we want to introduce some new concepts,
Bernoulli and Binomial trials, which are
repetitions of exactly the same experiments with
two possible outcomes.
In this case, we repeat the experiment of tossing
a fair coin 3 times, each time with 50% chance
of getting head and 50% chance of getting tail.
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Bernoulli and Binomial Trials
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Bernoulli Trials:
 An
experiment who has only two outcomes,
and. E.g., tossing a fair coin (head 50%, tail
50%); tossing a biased coin (head 70%, tail
30%); rolling a fair die and getting a 3 or more
(yes 4/6, no 2/6)
Bernoulli and Binomial Trials
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Binomial Trials:
 Repeating
Bernoulli trials for a number of
times, each repetition has the same possible
outcomes
 The probability of each outcome is consistent
for all trials.
Coin Toss Example
That is a Binomial experiment, or we say
the (discrete) random variable X follows
a Binomial distribution.
 For Binomial distribution, the outcomes
can be summarized with a pdf that does
not have to look like a table, but like a
function instead.
 Use our knowledge:
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Binomial Experiment
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An experiment is said to be a binomial
experiment if
 The
experiment consists of a sequence of n
identical trials
 Two outcomes (success/failure) are possible
on each trial.
 The probability of a success, p, does not
change from trial to trial.
 The trials are independent.
Binomial random variable
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Binomial random variable is a random variable
that describes the outcomes of a binomial
experiment.
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Example:
 1. Tossing a fair coin 100 times.
 2. Tossing a biased coin 100 times.
 3. Rolling a fair die 100 times and record the
numbers.
 4. Rolling a fair die 100 times and record whether the
outcome is even or odd.
 5. Rolling a snow ball on a ground covered with snow
and record whether it could pass a given distance.
Binomial random variable
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If a random variable describes the
outcome of a binomial experiment, we can
also say, this random variable follows a
binomial distribution, or this random
variable is binomially distributed.
A few words on probability
distribution
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A probability distribution is an approximation
to real life phenomenon.
It usually provides a functional relationship
between the possible values in the sample
space and their probabilities.
A probability distribution is always
characterized by parameters.
Therefore, knowing a probability distribution
means knowing its functional form and its
parameters.
Back to binomial distribution
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The functional form:
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The parameters: n and p.
Coin Toss Example
There are easier ways to find the expected
value and variance of a Binomial random
variable.
 If X~BIN(n,p)
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 E(X)=np
 Var(X)=np(1-p)
 In
this case, n=3, p=0.5, so
E(X)=np=3*0.5=1.5 and
Var(X)=3*0.5*0.5=0.75
More questions on coin tossing
What is the probability that we see at least
2 heads?
 That means the probability of seeing either
2 heads or 3 heads.
 P(X=2)+P(X=3)
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Another example
A player is shooting at a target 200 meters
away. There is 80% chance that he can hit
the target each time. He took 15 shots
within 10 minutes.
 A. How many times do you expect him to
hit the target? Also, find the standard
deviation of the number of times he hits
the target.
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Shooting example
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B. What is the chance that he missed
three times?
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C. What is the chance that he missed
more than 5 times?
Shooting example
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If the player pays $25 to play the game
gets a reward of $10 for each hit, what is
the expected amount of money he gets for
playing the game?
A more difficult example
Two players, A and B are playing a game.
A will roll a fair die and he wins if the
number is greater than 4. They repeat the
game 10 times.
 A. Let X be the number of games won by
B, find E(X) and Var(X).
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Card Game example
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B. What is the probability that B won at
least 4 games?
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What is the probability that A won more
than 7 games?
Card Game Example
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If A pays B $3 if B wins and B pays A $4 if
A wins, is this a fair game? (a fair game
means the expected payout from the
game should be zero).
More on E(X) and Var(X)
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We mentioned before that the expected
values have the following property:
 E(X+c)=E(X)+c
 E(aX)=aE(X)
 E(aX+c)=aE(X)+c
 E(aX+bY)=aE(X)+bE(Y)
More on E(X) and Var(X)
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Also, variances have similar properties:
 Var(X+b)=Var(X)
 Var(aX)=(a^2)Var(X)
 Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y).
If X and Y are
independent, Var(X+Y)=Var(X)+Var(Y).
 Var(aX+bY)=(a^2)Var(X)+(b^2)Var(Y)+2abCov(X,Y).
 If X and Y are independent,
Var(aX+bY)=(a^2)Var(X)+(b^2)Var(Y).
More on E(X) and Var(X)
*** The above properties are only for linear
transformations.
 If we have, for example, y=2*sqrt(X), the
above properties can not be used.
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An example on E(X) and Var(X)
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A biologist is conducting a research on the
temperature needed for chickens to be
hatched. His lab results are summarized
as the following,
Temperature
(C)
42
45
48
51
54
P(Hatching)
10%
15%
25%
35%
15%
An example on E(X) and Var(X)
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What is the mean and variance for the
temperature of hataching?
An example on E(X) and Var(X)
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The researcher’s lab assistant just found
out that the thermometer was
malfunctioning when the measures were
taken. All the temperatures on record are
5 degrees lower than they should be. Shall
the researcher re-do the experiment or do
something else to make it up?
An example on E(X) and Var(X)
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The researcher wants to submit his results
to apply for some grants. But the grant
committee requires that the temperature
should be recorded in terms of Fahrenheit
instead of Celsius. What should the
researcher do to update his data and
results.