Part V: Continuous Random Variables

http://rchsbowman.wordpress.com/2009/11/29/statistics-notes-%E2%80%93-properties-of-normal-distribution-2/

Chapter 23: Probability Density Functions

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

Comparison of Discrete vs. Continuous (Examples)

Discrete ContinuousCounting: 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.

Comparison of mass vs. densityMass (probability

mass function, PMF)Density (probability density

function, PDF)0 ≤ pX(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

Example 1 (class)Let x be a continuous random variable with density:

a) What is P(0 ≤ X ≤ 3)?b) Determine the CDF.c) Graph the density.d) Graph the CDF.e) Using the CDF, calculate

P(0 ≤ X ≤ 3), P(2.5 ≤ X ≤ 3)

Example 1 (cont.)

-1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

f(x)

-1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

F(x)

Example 2

Let X be a continuous function with CDF as follows

What is the density?

Comparison of CDFsDiscrete Continuous

Functiongraph Step function with

jumps of the same size as the mass

continuous

graph Range: 0 ≤ X ≤ 1 Range: 0 ≤ X ≤ 1

Example 3

Suppose a random variable X has a density given by:

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

Example 4Suppose a random variable X has the following density:

a) Find the CDF.b) Graph the density.c) Graph the CDF.

Example 4 (cont.)

-1 0 1 2 3 4 50

0.20.40.60.8

1

x

f(x)

-1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x

F(x)

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.

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.10.20.30.40.50.60.70.80.9

1

Chapter 24: Joint Densities

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

Probability for two continuous r.v.

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

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)

Example: FPF (Cont)

-10 0 10 20 30 40

-10

0

10

20

30

40

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 isfX,Y(x,y) = 10e-(2x+5y), x, y > 0and fX,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.

Example 2d

t

t

Example 2e

y = x

Example 2e

y = x

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.

Example 4 (class)

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

a) Verify that this is a valid joint density.b) Find the joint CDF.c) From the joint CDF calculated in a),

determine the density (which should be what is given above).

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 fX(x)?b) What is fY(y)?

2

X,Y

6(x y ) 0 x 1,0 y 1

f (x,y) 50 else

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 fX(x)?b) What is fY(y)?

X,Y

24xy 0 x 1,0 y 1,x y 1f (x,y)

0 else

Chapter 25: Independent

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

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

Example: Independent R.V.’sA 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?

2

X,Y

6(x y ) 0 x 1,0 y 1

f (x,y) 50 else

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, fY(y) = 5e-5y, y > 0and fX(x) = fY(y) = 0 otherwise.

What is fX,Y(x,y)?

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?

X,Y

24xy 0 x 1,0 y 1,x y 1f (x,y)

0 else

Chapter 26: Conditional Distributions

Q : What is conditional probability?A : maybe, maybe not.

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

Example: Conditional PDF (class)Suppose a random variables X and Y have a joint density given by:

a) Calculate the conditional density of X when Y = y where 0 < y < 1.

b) Verify that this function is a density.c) What is the conditional probability that X is

between -1 and 0.5 when we know that Y = 0.6.d) Are X and Y independent? (Show using

conditional densities.)

Chapter 27: Expected values

http://www.qualitydigest.com/inside/quality-insider-article/problems-skewness-and-kurtosis-part-one.html#

Comparison of Expected ValuesDiscrete Continuous

Example: Expected Value (class)

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

X

1 3x 0 x 2

f (x) 8 80 else

X

2 8 x 8.5f (x)

0 else

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:

Chapter 28: Functions, Variance

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

Comparison of Functions, VariancesDiscrete Continuous

Function (general)

Function (X2)

Variance Var(X) = (X2) – ((X))2 Var(X) = (X2) – ((X))2

SD

Example: Expected Value - function (class)

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

X

1 3x 0 x 2

f (x) 8 80 else

X

2 8 x 8.5f (x)

0 else

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 (X2) in each of the following situations:

Example: Variance (class)

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

X

1 3x 0 x 2

f (x) 8 80 else

X

2 8 x 8.5f (x)

0 else

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:

Friendly Facts about Continuous Random Variables - 1

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

(a1X1 + … + anXn) = 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))

Friendly Facts about Continuous Random Variables - 2

• Theorem 28.21: Variances of a linear sum of two or more independent continuous random variables:

Var(a1X1 + … + anXn) =Var(X1) + … + Var(Xn) • Corollary 28.22: Variance of a linear function

of continuous random variables:Var(aX + b) = a2Var(X)

Chapter 29: Summary and Review

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

Example: percentileLet x be a continuous random variable with density:

a) What is the 99th percentile?b) What is the median?