Chapter 2. Random Variables

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# Chapter 2. Random Variables - PowerPoint PPT Presentation

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

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

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)
• 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)
• 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 :

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

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)
• Example 1 : Machine Breakdowns
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)
• Probability Density Function (p.d.f.)
• Probabilistic properties of a continuous random variable

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

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)
• Cumulative Distribution Function
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)
• Example 1 (discrete random variable)
• The expected repair cost is
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)
• 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)
• 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 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)

Two distribution with identical mean values but different variances

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)
• 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)
• Example 14
• Upper quartile :
• Lower quartile :
• Interquartile range :
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)
• Joint Cumulative Distribution Function
• Discrete
• Continuous
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)
• Joint p.m.f
• Joint cumulative distribution function
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)
• Example 19
• Marginal p.m.f of X
• Marginal p.m.f of Y
Example 20: (a jointly continuous case)
• Joint pdf:
• Marginal pdf’s of X and Y:
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)
• Example 19
• Marginal probability distribution of Y
• Conditional distribution of X
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)
• 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)
• Example 19 (Air conditioner maintenance)
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)
• Example 19: (Air conditioner maintenance)

where

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

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)
• 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)
• For example, x=12 corresponds to a standardized score of y=(4ⅹ12)+20=68
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)
• 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)
• 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)
• Example 21
• The standardized scores of the two tests are
• The final score is
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)
• 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

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)
• CDF methd
• The p.d.f of Y is obtained by differentiation to be
• Notice that

Given and ,what is ?

If are all its real roots, that is,

then

where

Example: determine

-> one root is possible in