Part V: Continuous Random Variables - PowerPoint Presentation

Slide1

Part V: Continuous Random Variables

http://

rchsbowman.wordpress.com/2009/11/29

/

statistics-notes-%E2%80%93-properties-of-normal-distribution-2/Slide2

Chapter 23: Probability Density Functions

http://divisbyzero.com/2009/12/02/an-applet-illustrating-a-continuous-nowhere-differentiable-function//Slide3

Comparison of Discrete vs. Continuous (Examples)

Discrete

Continuous

Counting: defects, hits, die values, coin heads/tails, people, card arrangements, trials until success, etc.

Lifetimes, waiting times, height, weight, length, proportions, areas, volumes, physical quantities, etc.Slide4

Comparison of mass vs. density

Mass (probability mass function, PMF)

Density (probability density function, PDF)

0 ≤

p

X

(x)

≤ 1

0 ≤

fX(x)P(0 ≤ X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)P(X ≤ 3) ≠ P(X < 3) when P(X = 3) ≠ 0 P(X ≤ 3) = P(X < 3) since P(X = 3) = 0 always

Mass (probability mass function, PMF)

Density (probability density function, PDF)

0 ≤

p

X

(x)

≤ 1

0 ≤

f

X

(x)

P(0

≤ X

≤ 2)

= P(X = 0) + P(X = 1) + P(X = 2)

P(X

≤ 3) ≠ P(X < 3)

when P(X = 3) ≠ 0

P(X

≤ 3) = P(X < 3)

since P(X = 3) = 0 always

Slide5

Example 1 (class)

Let x be a continuous random variable with density:

What is P(0 ≤ X ≤ 3)?

Determine the CDF.

Graph the density.

Graph the CDF.

Using the CDF, calculate

P(0 ≤ X ≤ 3), P(2.5 ≤ X ≤ 3)

 Slide6

Example 1 (cont.)Slide7

Example 2

Let X be a continuous function with CDF as follows

What is the density?

 Slide8

Comparison of CDFs

Discrete

Continuous

Function

graph

Step

function with jumps of the same size as the mass

continuous

graph

Range: 0 ≤ X ≤ 1

Range: 0 ≤ X ≤ 1

Discrete

Continuous

Function

graph

Step

function with jumps of the same size as the mass

continuous

graph

Range: 0 ≤ X ≤ 1

Range: 0 ≤ X ≤ 1 Slide9

Example 3

Suppose a random variable X has a density given by:

Find the constant k so that this function is a valid density.

 Slide10

Example 4

Suppose a random variable X has the following density:

Find the CDF.

Graph the density.

Graph the CDF.

 Slide11

Example 4 (cont.)Slide12

Mixed R.V. – CDF

Let X denote a number selected at random from the interval (0,4), and let Y = min(X,3). Obtain the CDF of the random variable Y.Slide13

Chapter 24: Joint Densities

http://www.alexfb.com/cgi-bin/twiki/view/PtPhysics/WebHomeSlide14

Probability for two continuous r.v.

http://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspxSlide15

Example 1 (class)

A man invites his fiancée to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 am and 12 noon. If they arrive a random times during this period, what is the probability that they will meet within 10 minutes? (Hint: do this geometrically)Slide16

Example: FPF (Cont)Slide17

Example 2 (class)

Consider two electrical components, A and B, with respective lifetimes X and Y. Assume that a joint PDF of X and Y is fX,Y(x,y

) = 10e

-(2x+5y)

, x, y > 0

and

f

X,Y

(

x,y) = 0 otherwise.a) Verify that this is a legitimate density.b) What is the probability that A lasts less than 2 and B lasts less than 3?c) Determine the joint CDF.d) Determine the probability that both components are functioning at time t.e) Determine the probability that A is the first to fail.f) Determine the probability that B is the first to fail.Slide18

Example 2dSlide19

Example 2eSlide20

Example 2eSlide21

Example 3

Suppose a random variables X and Y have a joint density given by:

Find the constant k so that this function is a valid density.

 Slide22

Example 4 (class)

Suppose a random variables X and Y have a joint density given by:

Verify that this is a valid joint density.

Find the joint CDF.

From the joint CDF calculated in a), determine the density (which should be what is given above).

 Slide23

Example: Marginal density (class)

A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. The joint PDF is

a) What is

f

X

(x)?

b) What is

f

Y

(y)?Slide24

Example: Marginal density (homework)

A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is

a) What is

f

X

(x)?

b) What is

f

Y

(y)?Slide25

Chapter 25: Independent

Why’s everything got to be so intense with me? I’m trying to handle all this unpredictability In all probability

-- Long Shot, sung by Kelly Clarkson, from the album All I ever Wanted; song written by Katy Perry, Glen Ballard, Matt

ThiessenSlide26

Example: Independent R.V.’s

A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. The joint PDF is

Are X and Y independent?

 Slide27

Example: Independence

Consider two electrical components, A and B, with respective lifetimes X and Y with marginal shown densities below which are independent of each other. fX(x) = 2e

-2x

, x

>

0,

f

Y

(y

) = 5e-5y, y > 0 and fX(x) = fY(y) = 0 otherwise.What is fX,Y(x,y)?Slide28

Example: Independent R.V.’s (homework)

A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is

Are X and Y independent?Slide29

Chapter 26: Conditional Distributions

Q : What is

conditional

probability?

A : maybe, maybe not.

http://www.goodreads.com/book/show/4914583-f-in-examsSlide30

Example: Conditional PDF (class)

Suppose a random variables X and Y have a joint density

given by:

Calculate the conditional density of X

when

Y

= y

where 0 < y < 1.

Verify that this function is a density.What is the conditional probability that X is between -1 and 0.5 when we know that Y = 0.6.Are X and Y independent? (Show using conditional densities.) Slide31

Chapter 27: Expected values

http://

www.qualitydigest.com/inside/quality-insider-article

/

problems-skewness-and-kurtosis-part-one.html#Slide32

Comparison of Expected Values

Discrete

Continuous

Discrete

ContinuousSlide33

Example: Expected Value (class)

a) The following is the density of the magnitude X of a dynamic load on a bridge (in

newtons

)

b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

What is the expected value in each of the following situations:Slide34

Chapter 28: Functions, Variance

http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/Slide35

Comparison of Functions, Variances

Discrete

Continuous

Function (general)

Function

(X

2

)

Variance

Var

(X)

=

(X

2

) – (

(X))

2

Var(X)

=

(X

2

) – (

(X))

2

SD

Discrete

Continuous

Function (general)

Function

(X

2

)

Variance

SDSlide36

Example: Expected Value - function (class)

a) The following is the density of the magnitude X

of a dynamic load on a bridge (in

newtons

)

b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

What is

(X

2

) in each of the following situations: Slide37

Example: Variance (class)

a) The following is the density of the magnitude X of a dynamic load on a bridge (in

newtons

)

b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

What is the variance in each of the following situations:Slide38

Friendly Facts about Continuous Random Variables - 1

Theorem 28.18: Expected value of a linear sum of two or more continuous random variables:

(a

1

X

1

+ … +

a

n

Xn) = a1(X1) + … + an(Xn) Theorem 28.19: Expected value of the product of functions of independent continuous random variables:(g(X)h(Y)) = (g(X))(h(Y))  Slide39

Friendly Facts about Continuous Random Variables - 2

Theorem 28.21: Variances of a linear sum of two or more independent continuous random variables:Var(a1X

1

+ … +

a

n

X

n

) =

Var(X1) + … + Var(Xn) Corollary 28.22: Variance of a linear function of continuous random variables:Var(aX + b) = a2Var(X) Slide40

Chapter 29: Summary and Review

http://www.ux1.eiu.edu/~cfadd/1150/14Thermo/work.htmlSlide41

Example: percentile

Let x be a continuous random variable with density:

What is the 99

th

percentile?

What is the median?