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Chapter 2. Random Variables. 2.1 Discrete Random Variables 2.2 Continuous Random Variables 2.3 The Expectation of a Random Variable 2.4 The Variance of a Random Variable 2.5 Jointly Distributed Random Variables 2.6 Combinations and Functions of Random Variables. -3. -2. -1. 1. 0. 2. 3.

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chapter 2 random variables

Chapter 2. Random Variables

2.1 Discrete Random Variables

2.2 Continuous Random Variables

2.3 The Expectation of a Random Variable

2.4 The Variance of a Random Variable

2.5 Jointly Distributed Random Variables

2.6 Combinations and Functions of Random Variables

2 1 discrete random variable 2 1 1 definition of a random variable 1 2

-3

-2

-1

1

0

2

3

2.1 Discrete Random Variable2.1.1 Definition of a Random Variable (1/2)
  • Random variable
    • A numerical value to each outcome of a particular experiment
2 1 1 definition of a random variable 2 2
2.1.1 Definition of a Random Variable (2/2)
  • Example 1 : Machine Breakdowns
    • Sample space :
    • Each of these failures may be associated with a repair cost
    • State space :
    • Cost is a random variable : 50, 200, and 350
2 1 2 probability mass function 1 2
2.1.2 Probability Mass Function (1/2)
  • Probability Mass Function (p.m.f.)
    • A set of probability value assigned to each of the values taken by the discrete random variable
    • and
    • Probability :
2 1 2 probability mass function 1 25

0.5

0.3

0.2

50

200

350

2.1.2 Probability Mass Function (1/2)
  • Example 1 : Machine Breakdowns
    • P (cost=50)=0.3, P (cost=200)=0.2, P (cost=350)=0.5
    • 0.3 + 0.2 + 0.5 =1
2 1 3 cumulative distribution function 1 2

1.0

0.5

0.3

0

50

200

350

2.1.3 Cumulative Distribution Function (1/2)
  • Cumulative Distribution Function
    • Function :
    • Abbreviation : c.d.f
2 1 3 cumulative distribution function 2 2
2.1.3 Cumulative Distribution Function (2/2)
  • Example 1 : Machine Breakdowns
2 2 continuous random variables 2 2 1 example of continuous random variables 1 1
2.2 Continuous Random Variables2.2.1 Example of Continuous Random Variables (1/1)
  • Example 14 : Metal Cylinder Production
    • Suppose that the random variable is the diameter of a randomly chosen cylinder manufactured by the company. Since this random variable can take any value between 49.5 and 50.5, it is a continuous random variable.
2 2 2 probability density function 1 4
2.2.2 Probability Density Function (1/4)
  • Probability Density Function (p.d.f.)
    • Probabilistic properties of a continuous random variable
2 2 2 probability density function 2 4

49.5

50.5

2.2.2 Probability Density Function (2/4)
  • Example 14
    • Suppose that the diameter of a metal cylinder has a p.d.f
2 2 2 probability density function 4 4

49.8

50.1

49.5

50.5

2.2.2 Probability Density Function (4/4)
  • The probability that a metal cylinder has a diameter between 49.8 and 50.1 mm can be calculated to be
2 2 3 cumulative distribution function 1 3
2.2.3 Cumulative Distribution Function (1/3)
  • Cumulative Distribution Function
2 3 the expectation of a random variable 2 3 1 expectations of discrete random variables 1 2
2.3 The Expectation of a Random Variable2.3.1 Expectations of Discrete Random Variables (1/2)
  • Expectation of a discrete random variable with p.m.f
  • Expectation of a continuous random variable with p.d.f f(x)
  • The expected value of a random variable is also called the mean of the random variable
2 3 1 expectations of discrete random variables 2 2
2.3.1 Expectations of Discrete Random Variables (2/2)
  • Example 1 (discrete random variable)
    • The expected repair cost is
2 3 2 expectations of continuous random variables 1 2
2.3.2 Expectations of Continuous Random Variables (1/2)
  • Example 14 (continuous random variable)
    • The expected diameter of a metal cylinder is
    • Change of variable: y=x-50
2 3 2 expectations of continuous random variables 2 2
2.3.2 Expectations of Continuous Random Variables (2/2)
  • Symmetric Random Variables
    • If has a p.d.f that is symmetric about a point

so that

    • Then, (why?)
    • So that the expectation of the random variable is equal to the point of symmetry
2 3 3 medians of random variables 1 2
2.3.3 Medians of Random Variables (1/2)
  • Median
    • Information about the “middle” value of the random variable
  • Symmetric Random Variable
    • If a continuous random variable is symmetric about a point , then both the median and the expectation of the random variable are equal to
2 4 the variance of a random variable 2 4 1 definition and interpretation of variance 1 2
2.4 The variance of a Random Variable2.4.1 Definition and Interpretation of Variance (1/2)
  • Variance( )
    • A positive quantity that measures the spread of the distribution of the random variable about its mean value
    • Larger values of the variance indicate that the distribution is more spread out
    • Definition:
  • Standard Deviation
    • The positive square root of the variance
    • Denoted by
2 4 1 definition and interpretation of variance 2 2
2.4.1 Definition and Interpretation of Variance (2/2)

Two distribution with identical mean values but different variances

2 4 3 chebyshev s inequality 1 1
2.4.3 Chebyshev’s Inequality (1/1)
  • Chebyshev’s Inequality
    • If a random variable has a mean and a variance , then
    • For example, taking gives
2 4 4 quantiles of random variables 1 2
2.4.4 Quantiles of Random Variables (1/2)
  • Quantiles of Random variables
    • The th quantile of a random variable X
    • A probability of that the random variable takes a value less than the th quantile
  • Upper quartile
    • The 75th percentile of the distribution
  • Lower quartile
    • The 25th percentile of the distribution
  • Interquartile range
    • The distance between the two quartiles
2 4 4 quantiles of random variables 2 2
2.4.4 Quantiles of Random Variables (2/2)
  • Example 14
    • Upper quartile :
    • Lower quartile :
    • Interquartile range :
2 5 jointly distributed random variables 2 5 1 jointly distributed random variables 1 4
2.5 Jointly Distributed Random Variables2.5.1 Jointly Distributed Random Variables (1/4)
  • Joint Probability Distributions
    • Discrete
    • Continuous
2 5 1 jointly distributed random variables 2 4
2.5.1 Jointly Distributed Random Variables (2/4)
  • Joint Cumulative Distribution Function
    • Discrete
    • Continuous
2 5 1 jointly distributed random variables 3 4
2.5.1 Jointly Distributed Random Variables (3/4)
  • Example 19 : Air Conditioner Maintenance
    • A company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner
    • The random variable X, taking the values 1,2,3 and 4, is the service time in hours
    • The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units
2 5 1 jointly distributed random variables 4 4
2.5.1 Jointly Distributed Random Variables (4/4)
  • Joint p.m.f
  • Joint cumulative distribution function
2 5 2 marginal probability distributions 1 2
2.5.2 Marginal Probability Distributions (1/2)
  • Marginal probability distribution
    • Obtained by summing or integrating the joint probability distribution over the values of the other random variable
    • Discrete
    • Continuous
2 5 2 marginal probability distributions 2 2
2.5.2 Marginal Probability Distributions (2/2)
  • Example 19
    • Marginal p.m.f of X
    • Marginal p.m.f of Y
slide36
Example 20: (a jointly continuous case)
  • Joint pdf:
  • Marginal pdf’s of X and Y:
2 5 3 conditional probability distributions 1 2
2.5.3 Conditional Probability Distributions (1/2)
  • Conditional probability distributions
    • The probabilistic properties of the random variable X under the knowledge provided by the value of Y
    • Discrete
    • Continuous
    • The conditional probability distribution is a probability distribution.
2 5 3 conditional probability distributions 2 2
2.5.3 Conditional Probability Distributions (2/2)
  • Example 19
    • Marginal probability distribution of Y
    • Conditional distribution of X
2 5 4 independence and covariance 1 5
2.5.4 Independence and Covariance (1/5)
  • Two random variables X and Y are said to be independent if
    • Discrete
    • Continuous
    • How is this independency different from the independence among events?
2 5 4 independence and covariance 2 5
2.5.4 Independence and Covariance (2/5)
  • Covariance
    • May take any positive or negative numbers.
    • Independent random variables have a covariance of zero
    • What if the covariance is zero?
2 5 4 independence and covariance 3 5
2.5.4 Independence and Covariance (3/5)
  • Example 19 (Air conditioner maintenance)
2 5 4 independence and covariance 4 5
2.5.4 Independence and Covariance (4/5)
  • Correlation:
    • Values between -1 and 1, and independent random variables have a correlation of zero
2 5 4 independence and covariance 5 5
2.5.4 Independence and Covariance (5/5)
  • Example 19: (Air conditioner maintenance)
slide44
What if random variable X and Y have linear relationship, that is,

where

That is, Corr(X,Y)=1 if a>0; -1 if a<0.

2 6 combinations and functions of random variables 2 6 1 linear functions of random variables 1 4
2.6 Combinations and Functions of Random Variables2.6.1 Linear Functions of Random Variables (1/4)
  • Linear Functions of a Random Variable
    • If X is a random variable and for some numbers thenand
  • Standardization

-If a random variable has an expectation of and a variance of , has an expectation of zero and a variance of one.

2 6 1 linear functions of random variables 2 4
2.6.1 Linear Functions of Random Variables (2/4)
  • Example 21:Test Score Standardization
    • Suppose that the raw score X from a particular testing procedure are distributed between -5 and 20 with an expected value of 10 and a variance 7. In order to standardize the scores so that they lie between 0 and 100, the linear transformation is applied to the scores.
2 6 1 linear functions of random variables 3 4
2.6.1 Linear Functions of Random Variables (3/4)
  • For example, x=12 corresponds to a standardized score of y=(4ⅹ12)+20=68
2 6 1 linear functions of random variables 4 4
2.6.1 Linear Functions of Random Variables (4/4)
  • Sums of Random Variables
    • If and are two random variables, thenand
    • If and are independent, so that then
2 6 2 linear combinations of random variables 1 5
2.6.2 Linear Combinations of Random Variables (1/5)
  • Linear Combinations of Random Variables
    • If is a sequence of random variables and and are constants, then
    • If, in addition, the random variables are independent, then
2 6 2 linear combinations of random variables 2 5
2.6.2 Linear Combinations of Random Variables (2/5)
  • Averaging Independent Random Variables
    • Suppose that is a sequence of independent random variables with an expectation and a variance .
    • Let
    • Then and
    • What happened to the variance?
2 6 2 linear combinations of random variables 4 5
2.6.2 Linear Combinations of Random Variables (4/5)
  • Example 21
    • The standardized scores of the two tests are
    • The final score is
2 6 2 linear combinations of random variables 5 5
2.6.2 Linear Combinations of Random Variables (5/5)
  • The expected value of the final score is
  • The variance of the final score is
2 6 3 nonlinear functions of a random variable 1 3
2.6.3 Nonlinear Functions of a Random Variable (1/3)
  • Nonlinear function of a Random Variable
    • A nonlinear function of a random variable X is another random variable Y=g(X) for some nonlinear function g.
    • There are no general results that relate the expectation and variance of the random variable Y to the expectation and variance of the random variable X
2 6 3 nonlinear functions of a random variable 2 3

f(x)=1

E(x)=0.5

0

1

x

f(y)=1/y

E(y)=1.718

1.0

y

2.718

2.6.3 Nonlinear Functions of a Random Variable (2/3)
  • For example,
  • Consider
  • What is the pdf of Y?
2 6 3 nonlinear functions of a random variable 3 3
2.6.3 Nonlinear Functions of a Random Variable (3/3)
  • CDF methd
    • The p.d.f of Y is obtained by differentiation to be
  • Notice that
slide58
Determining the p.d.f. of nonlinear relationship between r.v.s:

Given and ,what is ?

If are all its real roots, that is,

then

where

slide59
Example: determine

-> one root is possible in