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 2.1 Discrete Random Variable 2.1.1 Definition of a Random Variable (1/2) Random variable A numerical value to each outcome of a particular experiment 2 1 3 -1 -2 -3

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 :

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 50 200 350 0.3 0.2 0.5 0.5 0.3 50 200 350 0.2

2.1.3 Cumulative Distribution Function (1/2) Abbreviation : c.d.f 1.0 0.5 0.3 50 200 350

2.1.3 Cumulative Distribution Function (2/2) Example 1 : Machine Breakdowns

2. 2 Continuous Random Variables 2. 2 2.2 Continuous Random Variables 2.2.1 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

2.2.2 Probability Density Function (2/4) Example 14 Suppose that the diameter of a metal cylinder has a p.d.f 49.5 50.5

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 49.5 50.5 49.8 50.1

2.2.3 Cumulative Distribution Function (1/3)

2.2.2 Probability Density Function (2/3) Example 14

2.2.2 Probability Density Function (3/3) 1 49.5 49.7 50.0 50.5

2. 3 The Expectation of a Random Variable 2. 3 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 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) 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

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

2. 4 The variance of a Random Variable 2. 4 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 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

2.4.2 Examples of Variance Calculations (1/1)

2.4.3 Chebyshev’s Inequality (1/1) If a random variable has a mean and a variance , then For example, taking gives

Proof

2.4.4 Quantiles of Random Variables (1/2) 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 Variables 2. 5 2.5 Jointly Distributed Random Variables 2.5.1 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) number of units X=service time 1 2 3 4 0.12 0.08 0.07 0.05 0.15 0.21 0.13 0.01 0.02 Joint p.m.f Joint cumulative distribution function

2.5.2 Marginal Probability Distributions (1/2) 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) 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) 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)

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 2.6 Combinations and Functions of Random Variables 2.6.1 Linear Functions of Random Variables (1/4) Linear Functions of a Random Variable If X is a random variable and for some numbers 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 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, then and If and are independent, so that then

Properties of

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

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) 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

2.6.3 Nonlinear Functions of a Random Variable (2/3) For example, Consider What is the pdf of Y? 1 x f(x)=1 E(x)=0.5 1.0 2.718 y f(y)=1/y E(y)=1.718

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