If we can reduce our desire,

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If we can reduce our desire, then all worries
that bother us will disappear.
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Random Variables and Distributions

• Distribution of a random variable
• Binomial and Poisson distributions
• Normal distributions

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What Is a Random Variable?

• The numerical outcome of a random circumstance is
  called a random variable.
• Eg. Toss a dice 1,2,3,4,5,6
• Height of a student
• A random variable (r.v.) assigns a number to each
  outcome of a random circumstance.
• Eg. Flip two coins the of heads

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Types of Random Variables

• A continuous random variable can take any value
  in one or more intervals.
• eg. Height, weight, age
• A discrete random variable can take one of a
  countable list of distinct values.
• eg. of courses currently taking

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Distribution of a Discrete R.V.

• X a discrete r.v.
• x a number X can take
• The probability distribution function (pdf) of X
  is
• P(X x)

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Example Birth Order of Children

• pdf Table 7.1 on page 163
• histogram of pdf Figure 7.1

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Important Features of a Distribution

• Overall pattern
• Central tendency mean
• Dispersion variance or standard deviation

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Calculating Mean Value

• X a discrete r.v.
• x1, x2, all possible X values
• pi is the probability X xi where i 1, 2,
• The mean of X is

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Variance Standard Deviation

• Notations as before
• Variance of X
• Standard deviation (sd) of X

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Example Birth Order of Children
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Bernoulli and Binomial Distributions

• A Bernoulli trial is a trial of a random
  experiment that has only two possible outcomes
  Success (S) and Failure (F). The notational
  convention is to let p P(S).
• Consider a fixed number n of identical (same
  P(S)), independent Bernoulli trials and let X be
  the number of successes in the n trials. Then X
  is called a binomial radon variable and its
  distribution is called a Binomial distribution
  with parameters n and p.
• Read the handout for bernoulli and binomial
  distributions.

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PDF of a Binomial R.V.

• p the probability of success in a trial
• n the of trials repeated independently
• X the of successes in the n trials
• For x 0, 1, 2, ,n,
• P(Xx)

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Mean Variance of a Binomial R.V.

• Notations as before
• Mean is
• Variance is

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Brief Minitab Instructions

• Minitab
• Calcgtgt Probability Distributionsgtgt Binomial
• Click probability , input constant and n, p,
  x
• Minitab Output
• Binomial with n 3 and p 0.29
• x P( X  x )
• 2 0.179133

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The Poisson Distribution

• a popular model for discrete events that occur
  rarely in time or space such as vehicle accident
  in a year
• The binomial r.v. X with tiny p and large n is
  approximately a Poisson r.v. for example, X
  the number of US drivers involved in a car
  accident in 2008
• Read the Poisson distribution handout.

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Brief Minitab Instructions

• Minitab
• Calcgtgt Probability Distributionsgtgt Poisson
• Click probability , input constant and l, x
• Minitab Output
• Poisson with mean 2.4
• x P( X  x )
• 1 0.217723

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Distribution of a Continuous R.V.

• The probability density function (pdf) for a
  continuous r.v. X is a curve such that
• P(a lt X ltb)
• the area under it over the interval a,b.

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Normal Distribution

• Its density curve is bell-shaped
• The distribution of a binomial r.v. with n8
• The distribution of a Poisson r.v. with l8
• Read the normal distribution handout.

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Standard Normal Distribution

• X a normal r.v. with mean m and standard
  deviation s
• Then is a normal r.v. with
  mean 0 and standard deviation 1 called a
  standard normal r.v.

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Brief Minitab Instructions

• Minitab
• Calcgtgt Probability Distributionsgtgt Normal Click
  what are needed
• Minitab Output
• Inverse Cumulative Distribution Function
• Normal with mean 0 and standard deviation 1
• P( X lt x ) x
• 0.95 1.64485
• Cumulative Distribution Function
• Normal with mean 0 and standard deviation 1
• x P( X lt x )
• 1.64485 0.950000

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Example Systolic Blood Pressure

• Let X be the systolic blood pressure. For the
  population of 18 to 74 year old males in US, X
  has a normal distribution with m 129 mm Hg and
  s 19.8 mm Hg.
• What is the proportion of men in the population
  with systolic blood pressures greater than 150 mm
  Hg?
• What is the 95-percentile of systolic blood
  pressure in the population?