How to Measure Uncertainty With Probability

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Chapter 7

• How to Measure Uncertainty With Probability

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Homework 13

• Read pages pages 454 - 470.
• LDI 7.20, 7.22 7.24
• Exercises 7.47, 7.48, 7.50, 7.59

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Random Variables

• A random variable is an uncertain numerical
  quantity whose value depends on the outcome of a
  random experiment. We can think of a random
  variable as a rule that assigns one and only one
  numerical value to each point of the sample space
  for a random experiment.

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Example

• If you play craps, the sum of the pips on the
  dice are the random variable. The random
  experiment is the rolling of the dice.

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• Random Process tossing a fair coin three
  times.
• There are eight possible individual outcomes in
  this sample space. Since the coin is assumed to
  be fair, the eight outcomes can be assumed to be
  equally likely (that is, the probability assigned
  to each individual outcome is 1/8).

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• 7.4.2 Rules of Probabilities
• To any event A, we assign a number P(A) called
  the probability of the event A.
• Assign a probability to each individual outcome,
  each being a number between 0 and 1, such that
  the sum of these individual probabilities is
  equal to 1, and,
• The probability of any event is the sum of the
  probabilities of the outcomes that make up that
  event.
• If the outcomes in the sample space are equally
  likely to occur, the probability of an event A is
  simply the proportion of outcomes in the sample
  space that make up the event A.

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Discrete Random Variable

• A discrete random variable can assume at most a
  finite or infinite but countable number of
  distinct values.
• Example Number of eggs a chicken lays in a day.
• Example Number of bombs dropped on a city.
• Example Number of casualties in a battle.

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Continuous Random Variable

• A continuous random variable can assume any value
  in an interval or collection of intervals. It
  always has an infinite number of possible values.
• Example Gallons of milk from a cow over its life
• Example Number of hours that CNN broadcasted the
  Iraq war with out interruption.
• Example Number of hours a battery will run a
  flashlight.

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• Give the values the random variable can take on
• X is the difference between the number of heads
  and number of tails obtained when a fair coin is
  tossed 3 times.
• Y is the product of the pips for the roll of 2
  fair dice.
• R is the time in minutes that this class lasts.

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• A discrete random variable can assume at most a
  finite or infinite but countable number of
  distinct values.
• A continuous random variable can assume any value
  in an interval or collection of intervals.

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Probability Distribution of a Discrete RV

• The probability distribution of a discrete random
  variable X is a table or rule that assigns a
  probability to each of the possible values of the
  discrete random variable X.

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Let X represent the number of people in an
apartment. Assume the maximum in a single
apartment is 7.
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Part (b)
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Lets Do It

• LDI 7.20, page 458

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The Mean of a Discrete Distribution

• The mean of a probability distribution is also
  called the expected value of the distribution.

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The Variance and Standard Deviation of a Discrete
Distribution

• The variance of a discrete probability
  distribution is
• The standard deviation is given by

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Good News!

• The TI-83 knows how to do these calculations. You
  simply enter the values of the random variable in
  L1 and the probabilities in L2 and do the
  following command
• 1-Var Stats L1,L2

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Apartments Revisited

• What is the expected value, the variance, and the
  standard deviation of the number of people per
  apartment?

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Lets Do It

• LDI 7.22

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Homework 14

• LDI 7.26, 7.27, 7.31
• Exercises 7.60, 7.61, 7.62, 7.65, 7.102, 7.103,
  7.104

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Combinations

• nCr represents the number of ways of selecting
  r items (without replacement) from a set of n
  distinct items where order of selection is not
  important.

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Bernoulli Variable

• If a random variable has exactly two possible
  outcome, success and failure and the probability
  of success remains fixed if the experiment is
  repeated under identical conditions, then the RV
  is dichotomous or Bernoulli.

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Lets Do It

• LDI 7.26

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Binomial Distribution

• A binomial random variable X is the total number
  of successes in n independent Bernoulli trials,
  on which each trial, the probability of success
  is p. We say X is B(n,p).
• Page 467 Blue box

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The Binomial Probability Distribution
Where p the probability of success in a single
trial q 1 p (probability of failure) n
number of independent trials x number of
successes in the n trials
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Lets Do It

• Page 470 LDI 7.27

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Mean and Standard Deviation of a Binomial RV
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Example

• A wart remover states it works on 95 of warts.
  If a total of 10 subjects are selected, what is
  the probability that 9 of the subjects will have
  their warts removed?

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Continuous Random Variables

• The probability distribution of a continuous
  variable X is a curve such that the area under
  the curve over an interval is equal to the
  probability that the random variable X is in the
  interval. The values of a continuous probability
  distribution must be at least 0 and the total
  area under the curve must be 1. The uniform and
  normal distributions we studied in chapter 6 were
  continuous.

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Approximating a Discrete RV with a Continuous One

• We can use the normal distribution to approximate
  the binomial when np 5 and np 5.
• If X is B(n, p) and np 5 and nq 5 then X can
  be approximated by

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Example

• A wart remover states it works on 95 of warts.
  If a total of 1000 subjects are selected, what is
  the probability that 900 of the subjects will
  have their warts removed?