Chapter 7, part I: Random Variables

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Chapter 7, part IRandom Variables

• Statistics 301
• Fall 2005
• October 27, 2005

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7.1 Random Variables

• A random variable (r.v.) is a numerical variable
  whose value depends on the outcome of a chance
  experiment.
• A r.v. is discrete if its set of possible values
  is a collection of isolated points on the number
  line.
• Almost always arises in connection with counting
• A r.v. is continuous if its set of possible
  values includes an entire interval on the number
  line.
• Almost always obtained from measurement

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7.1 Random Variables,Example

• State whether each of the following r.v.s is
  discrete or continuous
• Number of defective tires on a car
• Body temperature of a hospital patient
• Number of pages in a book
• Number of draws (with replacement) from a deck of
  cards until a heart is selected
• The lifetime, in hours, of a light bulb

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7.1 Random Variables,Example

• Starting at a particular time, each car entering
  an intersection is observed to note whether it
  turns left (L), turns right (R), or goes straight
  (S). The experiment terminates as soon as a car
  is observed to go straight. Let y denote the
  number of cars observed. What are possible
  values for y?

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7.1 Random Variables,Example (continued)
Notice that y is a discrete r.v. but the possible
number of values is not finite.
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7.2 Probability Distribution for Discrete R.V.s

• A probability distribution for a random variable
  is a model that describes the long-run behavior
  of the variable
• We might know things like the possible values
  for our variable but we might also be interested
  in the most common value and proportions and
  spreads
• A probability distribution provides this type of
  information

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7.2 Probability Distribution for Discrete R.V.s

• The probability distribution of a discrete random
  variable gives the probability associated with
  each possible value for the variable.
• Each probability is the limiting frequency in
  repeated trials.
• Commonly displayed in tables, histograms, or a
  formula.

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7.2 Probability Distribution for Discrete
R.V.sExample

• Suppose that a mail-order company has six
  telephone lines. Let X denote the number of
  lines in use at a specified time. The
  probability distribution for X is below.

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7.2 Probability Distribution for Discrete
R.V.sExample (continued)

• From this probability distribution, we can answer
  all sorts of questions!
• Whats the probability that at most three lines
  are in use?
• P(x0) P(x1) P(x2) P(x3)0.7
• Whats the probability that at least three lines
  are in use?
• P(x3) P(x4) P(x5) P(x6)0.55

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7.2 Probability Distribution for Discrete
R.V.sExample

• A couple wants three kids.
• Let x number of girls they have.
• If they have three kids, then they could have
• zero girls (x 0)
• one girl (x 1)
• two girls (x 2)
• three girls (x 3)

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7.2 Probability Distribution for Discrete
R.V.sExample (continued)

• Now I need the probabilities for each x.
• For x to equal 0, the couple needs to have three
  boys. The probability of this is
  (0.5)(0.5)(0.5)0.125.
• For x to equal 1, the couple needs to have two
  boys and the one girl. The probability of this
  is 3(0.5)(0.5)(0.5)0.375.
• For x to equal 2, the couple needs to have one
  boy and two girls. The probability of this is
  3(0.5)(0.5)(0.5)0.375.
• For x to equal 3, the couple needs to have three
  girls. The probability of this is
  (0.5)(0.5)(0.5)0.125.

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7.2 Probability Distribution for Discrete
R.V.sExample (continued)

• Putting these together, we can get the
  probability distribution for x.

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7.2 Probability Distribution for Discrete
R.V.sExample

• A discrete probability distribution in histogram
  form

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7.2 Probability Distribution for Discrete R.V.s

• Properties of Discrete Probability Distributions
• For every possible x value,
• 0 p(x) 1
• The sum of p(x) is 1.

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7.3 Probability Distribution for Continuous R.V.s

• A probability distribution for a continuous
  random variable, x, is specified by a
  mathematical function, f(x), and is called the
  density function.
• The graph of a density function is called a
  density curve.

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7.3 Probability Distribution for Continuous R.V.s

• Properties of density curves
• f(x) 0 (so the curve cannot dip below the
  x-axis)
• The total area under the curve is equal to 1.
• The probability that x falls in any particular
  interval is the area under the density curve and
  above the interval.

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7.3 Probability Distribution for Continuous
R.V.sExample

• Let x the amount of time (in minutes) that a
  particular Washington, D.C. commuter must wait
  for a Metro train. Suppose that the density
  curve is as pictured below

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7.3 Probability Distribution for Continuous
R.V.sExample (continued)

• That is, the density function is of the form
• It is easy to use this curve to calculate
  probabilities.
• Say like, P(8

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7.3 Probability Distribution for Continuous
R.V.sExample (continued)

• That is, we are interested in the area under the
  curve between 8 and 12
• Since this area is just a rectangle, we can find
  the area by (12-8)(0.05)0.20

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7.3 Probability Distribution for Continuous R.V.s

• This special density curve (horizontal line) has
  a name the uniform distribution
• Finding probabilities with uniform random
  variables will always be as easy as finding the
  area of a rectangle (base)(height)

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7.3 Probability Distribution for Continuous R.V.s

• Note the probability that a discrete r.v. lies
  in the interval between two limits depends on
  whether either limit is included in the interval
• i.e., P(x2) is not the same as P(x2)
• However, if we have a continuous r.v., it doesnt
  matter if we have or
• i.e., P(x2) P(x2)

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7.3 Probability Distribution for Continuous
R.V.sExample (revisited)

• Suppose we are interested in the waiting time for
  my morning commute and the commute home, that is
  z x y, the sum of the two waiting times.
• It can be shown that the density curve looks like
  this

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7.3 Probability Distribution for Continuous
R.V.sExample (continued)

• What is P(w
• 0.5

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7.3 Probability Distribution for Continuous
R.V.sExample (continued)

• What is P(w
• (0.5)(10-0)(0.025)0.125

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7.3 Probability Distribution for Continuous
R.V.sExample (continued)

• What is P(10
• 1-20.1250.75

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7.4 Mean and Standard Deviation of a R.V.

• We have already seen that the sample mean and
  sample standard deviation are numerical summaries
  for sample data.
• Similarly, the mean value and standard deviation
  of a random variable will describe the
  probability distribution.

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7.4 Mean and Standard Deviation of a R.V.

• The mean value of a r.v. x, denoted by µx,
  describes where the probability distribution is
  centered.
• The standard deviation of a r.v. sx, describes
  variability in the probability distribution.
• When sx is small, observed values for x tend to
  be close to the mean.
• When sx is large, there is more variability.

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7.4 Mean and Standard Deviation of a R.V.

• How do the means and standard deviations of the
  following density curves compare?

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7.4 Mean and Standard Deviation of a R.V.Example

• An appliance dealer sells three different models
  of upright freezers having 13.5, 15.9, and 19.1
  cubic feet of storage, respectively.
• Let x the amount of storage purchased by the
  next customer to buy a freezer.
• In a sample of 100 freezers recently purchased,
  the relative frequencies are shown below.

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7.4 Mean and Standard Deviation of a R.V.Example
(continued)

• The sample average of x for these 100 freezers is
  then the sum of 20 13.5 ft3, 50 15.9 ft3, and 30
  19.1 ft3 divided by 100.

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7.4 Mean and Standard Deviation of a R.V.

• Notice that the sample average is a weighted
  average of possible x values the weight of each
  value is the observed relative frequency.
• As sample size increases, each relative frequency
  approaches the corresponding probability.

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7.4 Mean and Standard Deviation of a R.V.

• The mean value of a discrete random variable is
  computed by first multiplying each possible x
  value by the probability of observing that value
  and then adding the resulting quantities.
• The term expected value is sometimes used in
  place of mean and E(x) is alternative notation
  for µx.

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7.4 Mean and Standard Deviation of a R.V.Example

• A chemical supply company currently has in stock
  100 lb. of a certain chemical, which it sells to
  customers in 5-lb lots. Let x the number of
  lots ordered by a randomly chosen customer. The
  probability distribution is as follows

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7.4 Mean and Standard Deviation of a R.V.Example
(continued)

• Calculate the mean value of x

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7.4 Mean and Standard Deviation of a R.V.

• Knowing only the mean of a distribution is only a
  partial summary.
• We also will probably want to know the spread of
  the distribution.
• The greater the spread implies that there will be
  more variability in a long sequence of observed x
  values.

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7.4 Mean and Standard Deviation of a R.V.

• The variance (the square of the standard
  deviation), denoted by s2, for a discrete r.v. is
  computed by first subtracting the mean from each
  possible x, then squaring each deviation and
  multiplying the result by the probability of the
  corresponding x value and finally adding these
  quantities.

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7.4 Mean and Standard Deviation of a R.V.Example
(revisited)

• Calculate the standard deviation of x
• Which implies that

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7.4 Mean and Standard Deviation of a R.V.Example
(revisited)

• Use your calculator to find the mean and standard
  deviation of a r.v.
• Put values of x in L1
• Put probabilities in L2
• 1VarStat L1, L2
• This weights the values with their corresponding
  probabilities

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7.4 Mean and Standard Deviation of a R.V.

• Mean and standard deviation when x is continuous
• Usually computationally intense
• These values will be given to you
• But they do have the same interpretation as for
  discrete

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7.4 Mean and Variation of Linear Functions and
Linear Combinations

• In some cases we might know the mean and standard
  deviation of one or more random variables
• We might also be interested in the behavior of
  some function of these variables

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7.4 Mean and Variation of Linear Functions and
Linear Combinations

• If x is a r.v. with mean µx and variance sx2 and
  if a and b are numerical constants, the random
  variable y defined by y abx is a linear
  function of x.
• The mean of y is µyabµx
• The variance of y is sy2b2 sx2.

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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample (revisited)

• Going back to the freezer example
• Suppose the price of the freezer depends on the
  size of the storage space, x, such that
  Price25x-8.5
• What is the mean value of the variable Price paid
  by the next customer?
• µyabµx-8.525(2.3)49

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7.4 Mean and Variation of Linear Functions and
Linear Combinations

• If x1, x2,, xn are random variables and a1,
  a2,, an are numerical constants, the random
  variable y defined by y a1 x1 a2 x2 an xn is
  a linear combination of the xis.
• We can easily compute the mean and standard
  deviation of y as long as the xis are
  independent.

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7.4 Mean and Variation of Linear Functions and
Linear Combinations

• If x1, x2,, xn are random variables with means
  µ1, µ2,, µn and variances s12, s22,, sn2,
  respectively, and if y is as previous then
• µy a1µ1 a2µ2 anµn
• sy2 a12 s12 a22 s22 an2 sn2

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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample

• Consider a small ferry that can accommodate cars
  and buses. The toll for cars is 3 and the toll
  for buses is 10. Let x and y denote the number
  of cars and buses, respectively, carried on a
  single trip. Cars and buses are accommodated on
  different levels of the ferry, so the number of
  buses accommodated on any trip is independent of
  the number of cars on the trip. Suppose that x
  and y have the following probability
  distributions

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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample (continued)
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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample (continued)

• Compute µx and standard deviation of x
• Check that you can find that µx2.8 and sx 1.29
• Compute µy and standard deviation of y
• Check that you can find that µy0.7 and sy 0.78

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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample (continued)

• Compute the mean and standard deviation for the
  total amount of money collected in tolls from
  cars (3x)
• µ3x 3µx 32.88.40
• s3x3sx 31.293.87
• Compute the mean and standard deviation for the
  total amount of money collected in tolls from
  buses (10y)
• µ10y 10µy 100.77.00
• s10y10sy 100.787.80

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7.4 Mean and Variation of Linear Functions and
Linear CombinationsExample (continued)

• Compute the mean and standard deviation for the
  total number of cars and buses (zxy)
• µz µx µy 2.80.73.5
• sz sqrt(sx2 sy2)sqrt(1.2920.782)1.51
• Compute the mean and standard deviation for the
  total amount of money collected in tolls
  (w3x10y)
• µw 3µy 10 µy 32.8100.715.40
• swsqrt(321.2921020.782)8.71