Chapter 3 Multidimensional random variables and their distribution 2

Iii. Continuous Random Variables

1. joint probability density

Define the Union distribution function of 3.3 (x, y) as f (x, y). If there is a non-negative function f (x, y), so that for any real number X, Y

F (x, y) = (3.12)

(X, Y) is a continuous random variable, and f (x, y) is the joint probability density of (x, y.

2. f (x, y) has the following properties:

Nature 1 F (x, y)Bytes0

Nature 2 = 1

Property 3 if the continuous point (x, y) of f (x, y) has

Property 4 if the random point (x, y) falls into a plane and is recorded as (x, y) d in Area D

P {(x, y) d} = (3.16)

Note: points in the f (x, y) Non-0 domain and the D public part have non-0 values.

P71 case 2

Example 3: (example 3.3 in the first version) set the joint probability density (x, y)

F (x, y) =

Evaluate the Union distribution function f (x, y) of (1) (x, y );

(2) P {x> 1}

(3) P {(x, y) d}, where D = {(x, y): x + y£.1 };

(4) P {X2BytesY}

Solution: note that the non-zero domain is H.

(1) When

,

Others

(2) P {x> 1} = 1-P {x1} = 1-fx (1) = 1-F (1, +)

=

(3) P {(x, y) d} =

=

=

=

=

(4) P {X2BytesY} =

=

=

=]

Note: The probability density is =

P {X2BytesY} = 1-