5.3 Random Variables

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5.3 Random Variables

• Random Variable
• Discrete Random Variables
• Continuous Random Variables
• Normal Distributions as Probability Distributions

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Random Variables
A random variable is a variable whose values are
numerical outcomes of a random experiment. That
is, we consider all the outcomes in a sample
space S and then associate a number with each
outcome
A random variable takes numerical values that
describe the outcomes of some chance process.
The probability distribution of a random
variable gives its possible values and their
probabilities. The probability histogram is a
graph showing this probability distribution
Example Consider tossing a fair coin 3
times. Define X the number of heads obtained in
3 tosses
X 0 TTT X 1 HTT THT TTH X 2 HHT HTH
THH X 3 HHH
X Values 0 1 2 3
Probability 1/8 3/8 3/8 1/8
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• The figure below shows how to get the probability
  distribution of X use a tree diagram! Each
  outcome has prob1/16 (HINT use the and rule
  to show this), and then use the or rule to show
  that P(X1) P(TTTH or TTHT or THTT or HTTT)
  etc)

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Discrete Random Variable
There are two main types of random variables
discrete and continuous. If we can find a way to
list all possible outcomes for a random variable
and assign probabilities to each one, we have a
discrete random variable.

• A discrete random variable X takes on a fixed set
  of possible values with gaps between. The
  probability distribution of a discrete random
  variable X lists the values xi and their
  probabilities pi
• Values of X x1 x2 x3
• P(X) p1 p2 p3
• The probabilities pi must satisfy two
  requirements
• Every probability pi is a number between 0 and
  1.
• The sum of the probabilities is 1.
• To find the probability of any event regarding X,
  add the probabilities pi of the particular values
  xi that make up the event.

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Continuous Random Variable
Discrete random variables commonly arise from
situations that involve counting something.
Situations that involve measuring something often
result in a continuous random variable.
A continuous random variable Y takes on its
values in an interval of numbers. The probability
distribution of Y is described by a density
curve. The probability of any event regarding Y
is the area under the density curve and above the
values of Y that make up the event.
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Continuous Probability Models
Suppose the random expt. is choose a number at
random between 0 and 1. We cannot assign
probabilities to each individual value because
there is an infinite interval of possible values,
so we model this with a density curve (that is
flat) that gives equal weight to all the numbers
between 0 and 1.
This continuous probability model assigns
probabilities as areas under the flat, uniform
density curve. The area under the curve and above
any range of values is the probability of an
outcome in that range. The total area under the
curve equals 1.
Example Find the probability of getting a random
number that is less than or equal to 0.5 OR
greater than 0.8.
Uniform distribution
P(X 0.5 or X gt 0.8) P(X 0.5) P(X gt 0.8)
0.5 0.2 0.7
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Normal Probability Models
There are many situations where the density curve
used to assign probabilities to intervals of
outcomes is Normal.

• Normal distributions can be thought of as
  probability models
• Probabilities can be assigned to intervals of
  outcomes using the Standard Normal probabilities
  in Table A.

We standardize normal data by calculating
z-scores so that any Normal curve N(m,s) can be
transformed into the standard Normal curve N(0,1).
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Normal Probability Models
Womens heights are Normally distributed with
mean 64.5 and standard deviation 2.5 in. What is
the probability, if we pick one woman at random,
that her height will be between 68 and 70 inches
P(68 lt X lt 70)? Because the woman is selected at
random, X is a random variable.
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5.4 Means and Variances of Random Variables

• The Mean of a Random Variable
• The Law of Large Numbers
• The Variance of a Random Variable
• HW problems for sections 5.3 and 5.4

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The Mean (Expected Value) of a Random Variable

• The mean of any discrete random variable is an
  average of the possible outcomes, with each
  outcome weighted by its probability. This
  reflects the fact that all outcomes may not be
  equally likely. The mean is also called the
  expected value.

Mean of a Discrete Random Variable Suppose that
X is a discrete random variable whose probability
distribution is Value x1 x2 x3 Proba
bility p1 p2 p3 To find the mean (expected
value) of X, multiply each possible value by its
probability, then add all the products so its
a weighted sum of the r.v.s values, the weights
being the probabilities.
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• Weve already discussed the mean of a density
  curve as being the balance point of the curve
  For a discrete r.v., well compute the mean (or
  expected value) as a weighted average of the
  values of X, the weights being the corresponding
  probabilities and in either case (discrete or
  continuous), the interpretation of the mean is as
  the long-run average value of X (in a large
  number of repetitions of the experiment giving
  rise to X).
• Example The mean of Hs in 4 tosses of a fair
  coin is computed as (1/16)0 (4/16)1
  (6/16)2 (4/16)3 (1/16)4 (32/16) 2.
• What is your mean winnings when you play the
  Pick 3 lottery? What does it mean?

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The Law of Large Numbers
Law of Large Numbers Essentially states that if
you sample from a population with mean m, then
the sample mean (x-bar) will approximate m for
large enough sample sizes. Or that m can be
thought of as the expected value (long-run
average value) of many independent observations
on the variable.
Draw independent observations at random from any
population with finite mean µ. Decide how
accurately you would like to estimate µ. The law
of large numbers says that as you increase the
number of observations, the sample mean of the
observed values gets closer and closer to the
mean µ of the population.
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Variance of a Random Variable

• Since we use the mean as the measure of center
  for a discrete random variable, well use the
  standard deviation as our measure of spread. The
  definition of the variance of a random variable
  is similar to the definition of the variance for
  a set of quantitative data.

Variance of a Discrete Random Variable Suppose
that X is a discrete random variable whose
probability distribution is Value x1 x2 x3
Probability p1 p2 p3 and that µX is
the mean of X. The variance of X is
To get the standard deviation of a random
variable, take the square root of the variance.
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HW Read sections 5.3 and 5.4 (up through p.
273, Law of Large Numbers) Dont worry much
about the computation of the variance of a
discrete random variable Work on Exercises
5.43, 5.48-5.55