Chapter5

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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/
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
Continuous
Counting: defects, hits, die Lifetimes, waiting times,
values, coin heads/tails,
height, weight, length,
people, card
proportions, areas,
arrangements, trials until
volumes, physical
success, etc.
quantities, etc.
Comparison of mass vs. density
Mass (probability
mass function, PMF)
0 ≤ pX(x) ≤ 1
Density (probability density
function, PDF)
0 ≤ fX(x)
∞
๐‘๐‘‹ ๐‘ฅ = 1
๐‘ฅ
๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ = 1
−∞
P(0 ≤ X ≤ 2) = P(X = 0)
๐‘ƒ
0
≤
๐‘‹
≤
2
=
+ P(X = 1) + P(X = 2)
P(X ≤ 3) ≠ P(X < 3)
when P(X = 3) ≠ 0
2
๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ
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:
1
๐‘“๐‘‹ ๐‘ฅ = 24 2๐‘ฅ + 3 1 ≤ ๐‘ฅ ≤ 4
0
๐‘’๐‘™๐‘ ๐‘’
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
f(x)
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5 1
x
0.8
F(x)
-1
0.6
0.4
0.2
0
-1
0
1
2
x
3
4
5
Example 2
Let X be a continuous function with CDF as
follows
0
๐‘ฅ<0
๐น๐‘‹ ๐‘ฅ = ๐‘ฅ 2 0 ≤ ๐‘ฅ ≤ 1
1
1<๐‘ฅ
What is the density?
Comparison of CDFs
Function
Discrete
Continuous
๐น๐‘‹ ๐‘Ž = ๐‘ƒ ๐‘‹ ≤ ๐‘Ž
๐น๐‘‹ ๐‘Ž = ๐‘ƒ ๐‘‹ ≤ ๐‘Ž
=
=
๐‘ƒ(๐‘‹ = ๐‘Ž)
๐‘ฅ≤๐‘Ž
graph
graph
Step function with
jumps of the same
size as the mass
Range: 0 ≤ X ≤ 1
๐‘Ž
๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ
−∞
continuous
Range: 0 ≤ X ≤ 1
Example 3
Suppose a random variable X has a density given
by:
4
๐‘˜๐‘ฅ
0
<
๐‘ฅ
<
4
๐‘“๐‘‹ ๐‘ฅ =
0
๐‘’๐‘™๐‘ ๐‘’
Find the constant k so that this function is a
valid density.
Example 4
Suppose a random variable X has the following
density:
1
0<๐‘ฅ<1
2
๐‘“๐‘‹ ๐‘ฅ = 1
1≤๐‘ฅ≤4
6
0
๐‘’๐‘™๐‘ ๐‘’
a) Find the CDF.
b) Graph the density.
c) Graph the CDF.
-1
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
x
1
0.8
F(x)
f(x)
Example 4 (cont.)
0.6
0.4
0.2
0
-1
0
1
2
x
3
4
5
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
0.8
0.6
0.4
0.2
0
-1
0
1
2
3
4
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)
40
30
20
10
0
-10
0
-10
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 is
fX,Y(x,y) = 10e-(2x+5y), x, y > 0
and 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:
๐‘˜๐‘ฅ๐‘ฆ 0 < ๐‘ฅ, ๐‘ฆ < 2
๐‘“๐‘‹,๐‘Œ ๐‘ฅ, ๐‘ฆ =
0
๐‘’๐‘™๐‘ ๐‘’
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:
๐‘ฅ + ๐‘ฆ 0 < ๐‘ฅ, ๐‘ฆ < 1
๐‘“๐‘‹,๐‘Œ ๐‘ฅ, ๐‘ฆ =
0
๐‘’๐‘™๐‘ ๐‘’
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
๏ƒฌ6
2
(x
๏€ซ
y
) 0 ๏‚ฃ x ๏‚ฃ 1,0 ๏‚ฃ y ๏‚ฃ 1
๏ƒฏ
fX,Y (x,y) ๏€ฝ ๏ƒญ 5
๏ƒฏ๏ƒฎ
0
else
a) What is fX(x)?
b) What is fY(y)?
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
๏ƒฌ24xy 0 ๏‚ฃ x ๏‚ฃ 1,0 ๏‚ฃ y ๏‚ฃ 1,x ๏€ซ y ๏‚ฃ 1
fX ,Y (x,y) ๏€ฝ ๏ƒญ
else
๏ƒฎ 0
a) What is fX(x)?
b) What is fY(y)?
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 Thiessen
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
๏ƒฌ6
2
๏ƒฏ (x ๏€ซ y ) 0 ๏‚ฃ x ๏‚ฃ 1,0 ๏‚ฃ y ๏‚ฃ 1
fX,Y (x,y) ๏€ฝ ๏ƒญ 5
๏ƒฏ๏ƒฎ
0
else
6
2
3 6 2
๐‘“๐‘‹ ๐‘ฅ = ๐‘ฅ + , ๐‘“๐‘Œ ๐‘ฆ = + ๐‘ฆ
5
5
5 5
Are X and Y independent?
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 > 0
and 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
๏ƒฌ24xy 0 ๏‚ฃ x ๏‚ฃ 1,0 ๏‚ฃ y ๏‚ฃ 1,x ๏€ซ y ๏‚ฃ 1
fX ,Y (x,y) ๏€ฝ ๏ƒญ
else
๏ƒฎ 0
Are X and Y independent?
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:
๐‘ฅ + ๐‘ฆ 0 < ๐‘ฅ, ๐‘ฆ < 1
๐‘“๐‘‹,๐‘Œ ๐‘ฅ, ๐‘ฆ =
0
๐‘’๐‘™๐‘ ๐‘’
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 Values
Discrete
Continuous
∞
๐”ผ ๐‘‹ =
๐‘ฅ๐‘๐‘‹ (๐‘ฅ)
๐‘ฅ
๐”ผ ๐‘‹ =
๐‘ฅ๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ
−∞
Example: Expected Value (class)
What is the expected value in each of the following
situations:
a) The following is the
density of the
magnitude X of a
dynamic load on a
bridge (in newtons)
๏ƒฌ1 3
๏ƒฏ ๏€ซ x 0๏‚ฃx๏‚ฃ2
fX (x) ๏€ฝ ๏ƒญ 8 8
๏ƒฏ๏ƒฎ 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.
๏ƒฌ 2 8 ๏€ผ x ๏€ผ 8.5
fX (x) ๏€ฝ ๏ƒญ
else
๏ƒฎ0
Chapter 28: Functions, Variance
http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/
Comparison of Functions, Variances
Discrete
Function
(general)
Continuous
๐”ผ ๐‘”(๐‘‹)
๐”ผ ๐‘”(๐‘‹)
=
=
∞
๐‘”(๐‘ฅ)๐‘๐‘‹ (๐‘ฅ)
−∞
๐‘ฅ
Function
2 =
๐”ผ
๐‘‹
(X2)
๐‘”(๐‘ฅ)๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ
∞
๐‘ฅ 2 ๐‘๐‘‹ (๐‘ฅ) ๐”ผ ๐‘‹ 2 =
๐‘ฅ 2 ๐‘“๐‘‹ ๐‘ฅ ๐‘‘๐‘ฅ
−∞
๐‘ฅ
Variance Var(X) = ๐”ผ(X2) – (๐”ผ(X))2 Var(X) = ๐”ผ(X2) – (๐”ผ(X))2
SD
๐œŽ๐‘‹ =
๐‘‰๐‘Ž๐‘Ÿ(๐‘‹)
๐œŽ๐‘‹ =
๐‘‰๐‘Ž๐‘Ÿ(๐‘‹)
Example: Expected Value - function
(class)
What is ๐”ผ(X2) in each of the following situations:
a) The following is the
density of the
magnitude X of a
dynamic load on a
bridge (in newtons)
๏ƒฌ1 3
๏ƒฏ ๏€ซ x 0๏‚ฃx๏‚ฃ2
fX (x) ๏€ฝ ๏ƒญ 8 8
๏ƒฏ๏ƒฎ 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.
๏ƒฌ 2 8 ๏€ผ x ๏€ผ 8.5
fX (x) ๏€ฝ ๏ƒญ
else
๏ƒฎ0
Example: Variance (class)
What is the variance in each of the following
situations:
a) The following is the
density of the
magnitude X of a
dynamic load on a
bridge (in newtons)
๏ƒฌ1 3
๏ƒฏ ๏€ซ x 0๏‚ฃx๏‚ฃ2
fX (x) ๏€ฝ ๏ƒญ 8 8
๏ƒฏ๏ƒฎ 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.
๏ƒฌ 2 8 ๏€ผ x ๏€ผ 8.5
fX (x) ๏€ฝ ๏ƒญ
else
๏ƒฎ0
Friendly Facts about Continuous
Random Variables - 1
• Theorem 28.18: Expected value of a linear
sum of two or more continuous random
variables:
๐”ผ(a1X1 + … + anXn) = a1๐”ผ(X1) + … + an๐”ผ(Xn)
• Theorem 28.19: Expected value of the product
of functions of independent continuous
random variables:
๐”ผ(g(X)h(Y)) = ๐”ผ(g(X))๐”ผ(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) =๐‘Ž12 Var(X1) + … + ๐‘Ž๐‘›2 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: percentile
Let x be a continuous random variable with
density:
1
๐‘“๐‘‹ ๐‘ฅ = 24 2๐‘ฅ + 3 1 ≤ ๐‘ฅ ≤ 4
0
๐‘’๐‘™๐‘ ๐‘’
0
๐‘ฅ<1
1 2
๐น๐‘‹ ๐‘ฅ =
๐‘ฅ + 3๐‘ฅ − 4 1 ≤ ๐‘ฅ ≤ 4
24
1
4<๐‘ฅ
a) What is the 99th percentile?
b) What is the median?
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