Transcript Random Variables 7.1 Discrete and Continuous Random Variables

Chapter 7: Random Variables
7.1
Discrete and Continuous
Random Variables
7.2
Means and Variances of
Random Variables
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Introduction
• A random variable is a function that associates a unique
numerical value with every outcome of an experiment.
The value of the random variable will vary from trial to
trial as the experiment is repeated.
• There are two types of random variables:
– Discrete
– Continuous
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Discrete Random Variables
• A discrete random variable is one which may take on
only a countable number of distinct values such as 0, 1,
2, 3, 4, ...
• If a random variable can take only a finite number of
distinct values, then it must be discrete.
• Examples of discrete random variables:
–
–
–
–
the number of children in a family,
the Friday night attendance at a cinema,
the number of patients in a doctor's surgery, and
the number of defective light bulbs in a box of ten.
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Continuous Random Variables
• A continuous random variable is one which
takes an infinite number of possible values.
Continuous random variables are usually
measurements.
• Examples of continuous random variables:
– height, weight, the amount of sugar in an
orange, the time required to run a mile.
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Examples
• A coin is tossed ten times. The random variable
X is the number of tails that are noted. X can
only take the values 0, 1, ..., 10, so X is a
discrete random variable.
• A light bulb is burned until it burns out. The
random variable Y is its lifetime in hours. Y can
take any positive real value, so Y is a
continuous random variable.
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Probability Distribution—
Discrete Random Variable
• The probability distribution of a discrete random
variable is a list of probabilities associated with each of
its possible values.
– More formally, the probability distribution of a discrete
random variable X is a function which gives the
probability p(xi) that the random variable equals xi, for
each value xi.
– See Example 7.1, p. 392
• The probability distribution satisfies the following
conditions:
1. 0  p( xi )  1
2.
 p( x )  1
i
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Examples
• Look at example 7.2, p. 394
• Problem 7.3, p. 396
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Probability Distribution—
Continuous Random Variable
• A density curve describes the probability distribution of a
continuous random variable.
– Recall density curves from Ch. 2 (especially pp. 78-83).
• The probability density function of a continuous random variable
is a function which can be integrated (find area under a curve, as
we have done using z-tables) to obtain the probability that the
random variable takes a value in a given interval.
• The probability distribution satisfies the following conditions:
– That the area under the curve must equal one.
– That the probability density function can never be negative:
f(x) > 0 for all x.
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Types of Continuous Random Variable
Probability Distributions (Density Functions)
• Uniform Distribution (Example 7.3, p. 398)
• Normal Distribution (Example 7.4, p. 400)
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Figure 7.5, p. 398: Uniform Distribution
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Figure 7.7, p. 400: Normal Distribution
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Practice
• Problem 7.6, p. 401
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Homework
• 7.2, p. 396
• 7.8, p. 402
• 7.10 and 7.11, p. 403
• Read through p. 403
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7.10, p. 403
Household
Family
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Mean of a Random Variable
• The mean of a random variable, also known as
its expected value, is the weighted average of all
the values that a random variable would
assume in the long run.
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Calculation of the Mean of a Discrete Random Variable
• To find the mean of a discrete random variable X,
multiply each possible value by its probability, then
add all the products. Just as probabilities are an
idealized description of long-run proportions, the
mean of a probability distribution describes the
long-run average outcome. For this statistic we use
the Greek letter µ:
 X   xi pi
• Problem 7.22, p. 411
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Variance of a Discrete Random Variable
• As we learned in earlier chapters, the mean of a distribution tells us
only part of the story—we also need a measure of spread.
• The variance of a discrete random variable is an average of the
squared deviation (X-µx)2 of the variable X from its mean µx.. As
with the mean, we use the weighted average in which each outcome
is weighted by its probability in order to take into account the
outcomes that are not equally likely. Recall that standard deviation
is simply the square root of variance.
 x2  ( x1   x ) 2 p1  ( x2   x ) 2 p2  ...  ( xk   x ) 2 pk
  (xi   x ) 2 pi
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Mean and Variance for a Continuous
Random Variable
• Both of these require calculus, and are not part
of the AP Stats curriculum.
– See notes at end of chapter if you are interested.
• We will focus on the mean and standard
deviations for discrete random variables.
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Problems
• Look over Example 7.7, p. 411.
• Now try Problems 7.25 and 7.26, p. 412.
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HW
• Additional Problems from 7.1
– 7.16, 7.17, 7.20, 7.21, pp. 405-407
• Reading:
– pp. 407-417
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Law of Large Numbers
• Would you expect a telephone survey to
provide a sample mean that is exactly the same
as the population mean?
• The law of large numbers tells us that if we take
a sample that is large enough, the mean (x-bar)
of the observed values will eventually approach
the mean (µ) of the population.
– See Example 7.8 and Figure 7.10, p. 414
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Figure 7.10, p. 414
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Another one, from Chapter 6
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Law of Large Numbers, Cont.
• The Law of Large Numbers says broadly
that the average results of many
independent observations are stable and
predictable (p. 415).
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Practice Problems
• Problems:
– 7.24, 7.29, p. 412
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HW
• Read through end of chapter.
– pp. 418-427
– Rules for Means and Variances
• Problems:
– 7.31, p. 417
– 7.42, p. 427
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Random Variables:
Rules for Means and Variances
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Rules for Means
(1) a bX  a  b X
(2)  X Y   X  Y
• For Rule (1), if you add a fixed value to each number in
a distribution, add this fixed value to the original mean
to get the new mean. If you multiply by a constant,
multiply the mean by the same constant.
• For Rule (2), simply add the means of two random
variables to get the mean of the new distribution.
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Rules for Variances
(1) 
(2) 
2
a  bX
2
X Y
b 
2
2
X
  
2
X
2
Y
Note: Rule 2 holds for independent random variables (see p. 421)
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Example 7.11, p. 421
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Practice Problems, pp. 425-426
• 7.36
• 7.37 and 7.38
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Combining Normal Random Variables
• Any linear combination of independent normal
random variables is also normally distributed.
– p. 424
• Find the mean and variance of the combined
random variables as above.
• See Example 7.14, p. 424
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Practice
Problem 7.45, p. 428
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Practice Problems
• 7.55, 7.57, 7.63
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