Download Jointly Distributed Random Variables 1 Discrete Random Variable

Jointly Distributed Random Variables
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Discrete Random Variable
• Joint Probability Mass Function
Joint probability mass function of two random variables X and Y
is
p(x, y) = P X = x, Y = y
X
P (X, Y ) ∈ A =
p(x, y)
(x,y)∈A
• Marginal Probability Mass Function
Marginal probability mass function of X and Y is respectively
X
PX (x) =
P X = x, Y = y
y∈all
PY (y) =
X
x∈all
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P X = x, Y = y
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Continuous Random Variable
• Joint Probability Density Function
f (x, y) is the joint probability density function of the continuous
random variables X and Y if
ZZ
P (X, Y ) ∈ A =
f (x, y)dxdy.
(x,y)∈A
Especially,
P a ≤ X ≤ b, c ≤ Y ≤ d =
Z bZ
a
d
f (x, y)dydx.
c
• Marginal Probability Density Function
Marginal probability density function of X and Y is respectively
Z ∞
fX (x) =
f (x, y)dy
−∞
Z ∞
fY (y) =
f (x, y)dx
−∞
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Independent Random Variables
X and Y are independent if
p(x, y) = PX (x) × PY (y)
f (x, y) = fX (x) × fY (y)
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X and Y are discrete
X and Y are continuous
Conditional Probability Distribution Functions
• X and Y are two continuous random variables with joint pdf
f (x, y). Then, conditional probability density function of Y given
X = a is
f (x, y)
fY |X=a Y a =
fX (a)
• If X and Y are discrete,
PY |X=a
P X = a, Y = y
Y a =
P X=a
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