Part V: Continuous Random Variables
description
Transcript of Part V: Continuous Random Variables
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Part V: Continuous Random Variables
http://rchsbowman.wordpress.com/2009/11/29/statistics-notes-%E2%80%93-properties-of-normal-distribution-2/
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Chapter 23: Probability Density Functions
http://divisbyzero.com/2009/12/02/an-applet-illustrating-a-continuous-nowhere-differentiable-function//
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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.
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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
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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)
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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)
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Example 2
Let X be a continuous function with CDF as follows
What is the density?
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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
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Example 3
Suppose a random variable X has a density given by:
Find the constant k so that this function is a valid density.
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Example 4Suppose a random variable X has the following density:
a) Find the CDF.b) Graph the density.c) Graph the CDF.
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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)
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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
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Chapter 24: Joint Densities
http://www.alexfb.com/cgi-bin/twiki/view/PtPhysics/WebHome
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Probability for two continuous r.v.
http://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx
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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)
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Example: FPF (Cont)
-10 0 10 20 30 40
-10
0
10
20
30
40
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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.
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Example 2d
t
t
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Example 2e
y = x
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Example 2e
y = x
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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.
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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).
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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
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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
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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
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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
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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)?
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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
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Chapter 26: Conditional Distributions
Q : What is conditional probability?A : maybe, maybe not.
http://www.goodreads.com/book/show/4914583-f-in-exams
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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.)
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Chapter 27: Expected values
http://www.qualitydigest.com/inside/quality-insider-article/problems-skewness-and-kurtosis-part-one.html#
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Comparison of Expected ValuesDiscrete Continuous
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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:
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Chapter 28: Functions, Variance
http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/
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Comparison of Functions, VariancesDiscrete Continuous
Function (general)
Function (X2)
Variance Var(X) = (X2) – ((X))2 Var(X) = (X2) – ((X))2
SD
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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:
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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:
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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))
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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)
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Chapter 29: Summary and Review
http://www.ux1.eiu.edu/~cfadd/1150/14Thermo/work.html
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Example: percentileLet x be a continuous random variable with density:
a) What is the 99th percentile?b) What is the median?