Binomial random variables

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• Binomial random variables

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Coin Toss Example

• 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.

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Coin Toss Example

• 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 12/81.5
• Then we can find the variance of the 8 numbers,
  which is
• (3-1.5)23(2-1.5)23(1-1.5)2(0-1.5)2/80.
  75

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Coin Toss Example

• Another way is to find the pmf.
• 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)(2-1.5)2(
  3/8)(3-1.5)2(1/8)0.75

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Coin Toss Example

• Yet another way.
• 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

• 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)

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Bernoulli and Binomial Trials

• 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.

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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

• 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.

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Binomial random variable

• Binomial random variable is a random variable
  that describes the outcomes of a binomial
  experiment.
• 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.

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Binomial random variable

• 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.

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A few words on probability distribution

• 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.

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Back to binomial distribution

• The functional form
• The parameters n and p.

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Coin Toss Example

• There are easier ways to find the expected value
  and variance of a Binomial random variable.
• If XBIN(n,p)
• E(X)np
• Var(X)np(1-p)
• In this case, n3, p0.5, so E(X)np30.51.5
  and Var(X)30.50.50.75

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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(X2)P(X3)

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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

• B. What is the chance that he missed three times?
• C. What is the chance that he missed more than 5
  times?

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Shooting example

• 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?

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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

• B. What is the probability that B won at least 4
  games?
• What is the probability that A won more than 7
  games?

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Card Game Example

• 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).

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More on E(X) and Var(X)

• We mentioned before that the expected values have
  the following property
• E(Xc)E(X)c
• E(aX)aE(X)
• E(aXc)aE(X)c
• E(aXbY)aE(X)bE(Y)

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More on E(X) and Var(X)

• Also, variances have similar properties
• Var(Xb)Var(X)
• Var(aX)(a2)Var(X)
• Var(XY)Var(X)Var(Y)2Cov(X,Y). If X and Y are
  independent, Var(XY)Var(X)Var(Y).
• Var(aXbY)(a2)Var(X)(b2)Var(Y)2abCov(X,Y).
• If X and Y are independent, Var(aXbY)(a2)Var(X)
  (b2)Var(Y).

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More on E(X) and Var(X)

• The above properties are only for linear
  transformations.
• If we have, for example, y2sqrt(X), the above
  properties can not be used.

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An example on E(X) and Var(X)

• A biologist is conducting a research on the
  temperature needed for chickens to be hatched.
  His lab results are summarized as the following,

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An example on E(X) and Var(X)

• What is the mean and variance for the temperature
  of hataching?

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An example on E(X) and Var(X)

• The researchers 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?

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An example on E(X) and Var(X)

• 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.