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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

-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 :

0.3

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

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

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 (3/4)

- This is a valid p.d.f.

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

2.2.2 Probability Density Function (2/3)

- Example 14

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

2.3.3 Medians of Random Variables (2/2)

- Example 14

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)

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 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

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

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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