Chapter 2. Random Variables2.1 Discrete Random Variables2.2 Continuous Random Variables2.3 The Expectation of a Random Variable2.4 The Variance of a Random Variable2.5 Jointly Distributed Random Variables2.6 Combinations and Functions of Random Variables

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 BreakdownsSample space : Each of these failures may be associated with a repair costState 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 :

2.1.2 Probability Mass Function (1/2)Example 1 : Machine BreakdownsP (cost=50)=0.3, P (cost=200)=0.2, P (cost=350)=0.50.3 + 0.2 + 0.5 =1

502003500.30.20.5

2.1.3 Cumulative Distribution Function (1/2)Cumulative Distribution FunctionFunction : Abbreviation : c.d.f

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)Example 14 : Metal Cylinder ProductionSuppose 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

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

2.2.2 Probability Density Function (3/4)This is a valid p.d.f.

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

2.2.2 Probability Density Function (2/3)Example 14

2.2.2 Probability Density Function (3/3)49.550.549.750.01

2.3 The Expectation of a Random Variable 2.3.1 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 VariablesIf 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)MedianInformation about the middle value of the random variable

Symmetric Random VariableIf a continuous random variable is symmetric about a point , then both the median and the expectation of the random variable are equal to

2.3.3 Medians of Random Variables (2/2)Example 14

2.4 The variance of a Random Variable 2.4.1 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 valueLarger values of the variance indicate that the distribution is more spread out

Definition:

Standard DeviationThe positive square root of the varianceDenoted by

2.4.1 Definition and Interpretation of Variance (2/2)Two distribution with identical mean values but different variances

2.4.2 Examples of Variance Calculations (1/1)Example 1

Chebyshevs InequalityIf a random variable has a mean and a variance , then

For example, taking gives

2.4.3 Chebyshevs Inequality (1/1)

Proof

2.4.4 Quantiles of Random Variables (1/2)Quantiles of Random variablesThe th quantile of a random variable X

A probability of that the random variable takes a value less than the th quantileUpper quartileThe 75th percentile of the distributionLower quartileThe 25th percentile of the distributionInterquartile rangeThe 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 Variables 2.5.1 Jointly Distributed Random Variables (1/4)Joint Probability DistributionsDiscrete

Continuous

2.5.1 Jointly Distributed Random Variables (2/4)Joint Cumulative Distribution FunctionDiscrete

Continuous

2.5.1 Jointly Distributed Random Variables (3/4)Example 19 : Air Conditioner MaintenanceA company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient mannerThe 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

Y=number of unitsX=service time123410.120.080.070.0520.080.150.210.1330.010.010.020.07

2.5.2 Marginal Probability Distributions (1/2)Marginal probability distributionObtained by summing or integrating the joint probability distribution over the values of the other random variableDiscrete

Continuous

2.5.2 Marginal Probability Distributions (2/2)Example 19Marginal p.m.f of X

Marginal p.m.f of Y

Example 20: (a jointly continuous case)

Joint pdf:

Marginal pdfs of X and Y:

2.5.3 Conditional Probability Distributions (1/2)Conditional probability distributionsThe probabilistic properties of the random variable X under the knowledge provided by the value of YDiscrete

Continuous

The conditional probability distribution is a probability distribution.

2.5.3 Conditional Probability Distributions (2/2)Example 19Marginal 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 ifDiscrete

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 zeroWhat 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)

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

2.6 Combinations and Functions of Random Variables

2.6.1 Linear Functions of Random Variables (1/4)Linear Functions of a Random VariableIf X is a random variable and for some numbers then and

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 StandardizationSuppose 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=(412)+20=68

2.6.1 Linear Functions of Random Variables (4/4)Sums of Random VariablesIf and are two random variables, then and If and are independent, so that then

Properties of

2.6.2 Linear Combinations of Random Variables (1/5)Linear Combinations of Random VariablesIf 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 VariablesSuppose 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 (3/5)

2.6.2 Linear Combinations of Random Variables (4/5)Example 21The 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 VariableA 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)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

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

Example: determine

-> one root is possible in

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