Download Tutorial 4 - Probability Density 2

Tutorial 4
Probability Density 2
ENGG2450A Tutors
The Chinese University of Hong Kong
13 February 2017
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The Uniform Distribution
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The uniform distribution
Definition
Uniform distribution. We say a continuous random variable X has uniform
distribution on [α, β] if it has the following probability density function,
1
β−α , for α < x < β,
f (x) =
0,
otherwise.
Mean of the uniform distribution is given by
µ=
β+α
.
2
Variance of the uniform distribution is given by
σ2 =
1
(β − α)2 .
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Joint Distribution
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Joint p.d.f of 2 continuous random variables
Definition
Joint p.d.f.. The joint probability density function of random variables X1
and X2 is a function f (x1 , x2 ) that possesses the properties
2
f (x1 , x2 ) ≥ 0;
R +∞ R +∞
−∞ −∞ f (x1 , x2 )dx1 dx2 = 1;
3
P(a1 ≤ X1 ≤ b1 , a1 ≤ X2 ≤ b2 ) =
1
R b2 R b1
a2
a1
f (x1 , x2 )dx1 dx2 .
Definition
Marginal p.d.f.. The maginal probabiliy density functions of random variables
X1 and X2 are respectively
Z +∞
Z +∞
f1 (x1 ) =
f (x1 , x2 )dx2 , and f2 (x2 ) =
f (x1 , x2 )dx1 .
−∞
−∞
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Joint cumulative distribution function and conditional p.d.f.
Definition
Joint cumulative distribution function. The joint cumulative distribution
function is
Z x2 Z x1
F (x1 , x2 ) =
f (u, v )dudv = P(X1 ≤ x1 , X2 ≤ x2 ),
−∞
−∞
so
f (x1 , x2 ) =
∂ 2 F (x1 , x2 )
.
∂x1 ∂x2
Definition
Conditional p.d.f.. The conditional probabiliy density of random variable X1
1 ,x2 )
being x1 given that random variable X2 being x2 is f (x1 |x2 ) = f f(x2 (x
2)
provided that f2 (x2 ) 6= 0.
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Independent random variable
Definition
Independence. Random variables X1 and X2 with p.d.f. f1 (x1 ) and f2 (x2 )
respectively are independent if and only if
f (x1 , x2 ) = f1 (x1 )f2 (x2 ).
If X1 and X2 are independent, we have
f1 (x1 |x2 ) =
f1 (x1 )f2 (x2 )
f (x1 , x2 )
=
= f1 (x1 ).
f2 (x2 )
f2 (x2 )
f2 (x2 |x1 ) =
f (x1 , x2 )
f1 (x1 )f2 (x2 )
=
= f2 (x2 ).
f1 (x1 )
f1 (x1 )
Similarly,
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