Transcript Chapters 25, 26, and 27

Chapter 25: 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:
𝑘𝑥𝑦 0 < 𝑥, 𝑦 < 2
𝑓𝑋,𝑌 𝑥, 𝑦 =
0
𝑒𝑙𝑠𝑒
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:
𝑥 + 𝑦 0 < 𝑥, 𝑦 < 1
𝑓𝑋,𝑌 𝑥, 𝑦 =
0
𝑒𝑙𝑠𝑒
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
6
2
(x

y
) 0  x  1,0  y  1

fX,Y (x,y)   5

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
24xy 0  x  1,0  y  1,x  y  1
fX ,Y (x,y)  
else
 0
a) What is fX(x)?
b) What is fY(y)?
Chapter 26: 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
6
2
 (x  y ) 0  x  1,0  y  1
fX,Y (x,y)   5

0
else
6
2
3 6 2
𝑓𝑋 𝑥 = 𝑥 + , 𝑓𝑌 𝑦 = + 𝑦
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
24xy 0  x  1,0  y  1,x  y  1
fX ,Y (x,y)  
else
 0
Are X and Y independent?
Chapter 27: 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:
𝑥 + 𝑦 0 < 𝑥, 𝑦 < 1
𝑓𝑋,𝑌 𝑥, 𝑦 =
0
𝑒𝑙𝑠𝑒
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.)