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Unformatted text preview: uppose f (x , y ) is deﬁned
1 0.15 0.10
2 0.25 0.30 0.20 What value of C makes it a valid
Find P (X > Y )
Find P (X + Y = 2)
What are fx and fy ?
What are E (X ) and E (Y )?
What is E (X + Y )?
Are X and Y independent? Culpepper, SA STAT 400: Statistics and Probability I 4.1: Distributions of two random variables
4.2: The Correlation Coeﬃcient Discrete Deﬁnitions
Continuous Examples Continuous Bivariate Distributions, Deﬁnitions
We can replace sums for integrals if X and Y are instead
continuous random variables.
A continuous joint p.d.f., f (x , y ), must satisfy the following
f (x , y ) ≥ 0 (f (x , y ) = 0 if (x , y ) is outside the support for X
and Y )
f (x , y ) dxdy = 1
P [(X , Y ) ∈ A] =
f (x , y ) dxdy
A The marginal p.d.f.s are deﬁned as,
∞ f (x , y ) dy , x ∈ S1 (4) f (x , y ) dx , y ∈ S2 f1 (x ) = (5) −∞
∞ f2 (y ) =
−∞ X and Y are independent if f (x , y ) = f1 (x ) f2 (y )
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