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Section 08
ο‚‘
𝑓 π‘₯, 𝑦 = 𝑃 𝑋 = π‘₯ ∩ π‘Œ = 𝑦
 defined over a two-dimensional region
ο‚‘
Discrete:

ο‚‘
𝑦𝑓
π‘₯, 𝑦 = 1
Continuous:

ο‚‘
π‘₯
∞
𝑓
βˆ’βˆž
π‘₯, 𝑦 𝑑𝑦 𝑑π‘₯ = 1
X and Y may be independent or dependent
ο‚‘
𝐹 π‘₯, 𝑦 = 𝑃[ 𝑋 ≀ π‘₯ ∩ π‘Œ ≀ 𝑦 ]
ο‚‘
Discrete:
π‘₯
𝑠=βˆ’βˆž
 𝐹 π‘₯, 𝑦 =
ο‚‘
𝑦
𝑑=βˆ’βˆž 𝑓(𝑠, 𝑑)
Continuous:
 𝐹 π‘₯, 𝑦 =

πœ•2
𝐹
πœ•π‘₯πœ•π‘¦
π‘₯
𝑦
𝑓
βˆ’βˆž βˆ’βˆž
𝑠, 𝑑 𝑑𝑑 𝑑𝑠
π‘₯, 𝑦 = 𝑓(π‘₯, 𝑦)
ο‚‘
If β„Ž(π‘₯, 𝑦) is a function of two variables, and X
and Y are jointly distributed random
variables, the expected value of β„Ž(π‘₯, 𝑦) is
 𝐸 β„Ž π‘₯, 𝑦
 𝐸 β„Ž π‘₯, 𝑦
=
=
continuous
β„Ž π‘₯, 𝑦 βˆ— 𝑓(π‘₯, 𝑦) for discrete
∞
β„Ž
βˆ’βˆž
π‘₯, 𝑦 βˆ— 𝑓 π‘₯, 𝑦 𝑑𝑦 𝑑π‘₯ for
ο‚‘
If X and Y have joint probability distribution
f(x,y) then the marginal distribution of X is
 𝑓𝑋 π‘₯ =
 𝑓𝑋 π‘₯ =
ο‚‘
𝑦 𝑓(π‘₯, 𝑦) for discrete
∞
𝑓 π‘₯, 𝑦 𝑑𝑦 for continuous
βˆ’βˆž
Think of it as β€œadding up” the probability for
all points where X is a certain value to get the
overall probability that X=x.
ο‚‘
Random variables X and Y are independent if:
 The probability space is rectangular AND
 𝑓 π‘₯, 𝑦 = 𝑓𝑋 π‘₯ βˆ— π‘“π‘Œ (𝑦)
ο‚‘
The second condition can also be determined
via 𝐹 π‘₯, 𝑦 = 𝐹𝑋 π‘₯ βˆ— πΉπ‘Œ (𝑦)
ο‚‘
Recall that 𝑃 𝐴 𝐡 =
𝑃(𝐴∩𝐡)
𝑃(𝐡)
 The same concept is used to find conditional
distributions
ο‚‘
𝑃 𝑋 π‘Œ = 𝑦 = 𝑓𝑋|π‘Œ (π‘₯|π‘Œ = 𝑦) =
𝑓(π‘₯,𝑦)
π‘“π‘Œ (𝑦)
ο‚‘
Expectation:
 𝐸 π‘Œπ‘‹=π‘₯ =
ο‚‘
𝑦 βˆ— π‘“π‘Œ|𝑋 𝑦 𝑋 = π‘₯ 𝑑𝑦
Variance:
 π‘‰π‘Žπ‘Ÿ π‘Œ 𝑋 = π‘₯ = 𝐸 π‘Œ 2 𝑋 = π‘₯ βˆ’ 𝐸 π‘Œ 𝑋 = π‘₯ 2
=
𝑦 2 βˆ— π‘“π‘Œ|𝑋 𝑦 𝑋 = π‘₯ 𝑑𝑦 βˆ’ [ 𝑦 βˆ— π‘“π‘Œ|𝑋 𝑦 𝑋 = π‘₯ 𝑑𝑦]2
ο‚‘
πΆπ‘œπ‘£ 𝑋, π‘Œ = 𝐸 π‘‹π‘Œ βˆ’ 𝐸 𝑋 βˆ— 𝐸(π‘Œ)
 If covariance is zero, X and Y are independent
ο‚‘
Covariance is used to find the variance of the
sum of X and Y
 π‘‰π‘Žπ‘Ÿ π‘Žπ‘‹ + π‘π‘Œ = π‘Ž2 π‘‰π‘Žπ‘Ÿ 𝑋 + 𝑏 2 π‘‰π‘Žπ‘Ÿ π‘Œ +
2π‘Žπ‘πΆπ‘œπ‘£(𝑋, π‘Œ)
ο‚‘
Coefficient of correlation:
 𝜌 𝑋, π‘Œ = πœŒπ‘‹,π‘Œ =
πΆπ‘œπ‘£(𝑋,π‘Œ)
πœŽπ‘‹ πœŽπ‘Œ
A device runs until either of two components
fails, at which point the device stops running.
The joint density function of the lifetimes of the
two components, both measured in hours, is
f(x, y) = (x+y)/8, for 0<x<2 and 0<y<2.
Calculate the probability that the device fails
during its first hour of operation
An insurance company insures a large number of drivers.
Let X be the random variable representing the
company’s losses under collision insurance, and let Y
represent the company’s losses under liability insurance.
X and Y have joint density function
2π‘₯ + 2 βˆ’ 𝑦
for 0 < π‘₯ < 1 and 0 < 𝑦 < 2
𝑓 π‘₯, 𝑦 =
4
0 otherwise
What is the probability that the total loss is at least 1?
A joint density function is given by
f(x,y) = kx, 0<x<1 & 0<y<1
Where k is a constant.
Calculate Cov(X,Y).
Let X and Y be continuous random variables
with joint density function
f(x, y) = (8/3)xy, 0 < x < 1, x < y < 2x
Calculate the covariance of X and Y.
Let X and Y be continuous random variables
with joint density function
f(x,y) = 15y, x^2 <= y <= x
Determine g, the marginal density function of
Y, including its support.
The distribution of Y, given X, is uniform on the
interval [0,X]. The marginal density of X is
f(x) = 2x, 0<x<1
Determine the conditional variance of X, given
Y = y.
An actuary analyzes a company’s annual
personal auto claims, M, and annual
commercial auto claims, N. The analysis reveals
that Var(M) = 1600, Var(N) = 900, and the
correlation between M and N is 0.64.
Calculate Var(M+N).
The joint probability density function of X and Y
is given by
f(x, y) = (x+y)/8 0<x<2 0<y<2
Calculate the variance of (X + Y)/2.
A policyholder has probability 0.7 of having no
claims, 0.2 of having exactly one claim, and 0.1
of having exactly two claims. Claim amounts
are uniformly distributed on the interval [0,60]
and are independent. The insurer covers 100%
of each claim.
Calculate the probability that the total benefit
paid to the policyholder is 48 or less.
X and Y have the joint cumulative distribution
function
F(x, y) = xy(x+y)/2,000,000 0<x<100; 0<y<100
Calculate Var(X).
A city with borders forming a square with sides of length
1 has its city hall located at the origin when a rectangular
coordinate system is imposed on the city so that two
sides of the square are on the positive axes. The density
function of the population is
f(x,y) = 1.5(x^2 + y^2), 0<x,y<1
A resident of the city can travel to the city hall only
along a route whose segments are parallel to the city
borders.
Calculate the expected value of the travel distance to
the city hall of a randomly chosen resident of the city.