# The joint probability density function of x and y is given by f(x y) = e^-(x+y)

the joint likelihood thickness purpose of X and Y is offered
by

f(x,y)={e-(x+y)
for X>0, y>0
and 0 in other places

A. discover limited thickness of X

B. discover limited thickness of Y

C. discover Conditional thickness of X offered Y

D. tend to be arbitrary factors X and Y independent? State the main reason
of one’s response.

E. Discover P(X<.5, y<.5) F. Discover P(X=.5, y<.5) the joint probability density function of X and Y is given by the joint probability density function of X and Y is given by f(x,y)={e-(x+y) for X>0, y>0
and 0 in other places f(x,y)={e-(x+y)
for X>0, y>0
and 0 elsewhere -(x+y)
A. discover limited thickness of X A. discover limited thickness of X
B. discover limited thickness of Y B. discover limited thickness of Y
C. discover Conditional thickness of X given Y C. discover Conditional thickness of X offered Y
D. tend to be arbitrary factors X and Y independent? State the main reason
of one’s response. D. tend to be arbitrary factors X and Y independent? State the main reason
of one’s response.
E. Discover P(X<.5, y<.5) E. Discover P(X<.5, y<.5) F. Discover P(X=.5, y<.5) F. Discover P(X=.5, y<.5)

General assistance

Ideas and explanation
The combined likelihood thickness purpose when it comes to constant arbitrary adjustable denote as
Often multiple dimension is taken for each person in the test area. In the event such as this you will have some arbitrary adjustable defined for a passing fancy likelihood area therefore we want to explore their particular combined circulation.

Principles

The combined likelihood thickness purpose fulfills here presumptions.

For almost any offered area of two-dimensional area,

Step by step

1 of 6

a)
The offered combined thickness purpose is,

Determine the limited circulation purpose of

Component a
The limited circulation purpose of is

The limited circulation purpose of is

Make use of the offered combined circulation purpose to determine the limited circulation purpose of

Step two of 6

b)
The offered combined thickness purpose is,

Determine the limited circulation purpose of

Component b
The limited circulation purpose of is

The limited circulation purpose of is

Make use of limited circulation features to determine the conditional thickness of

Step three of 6

c)
Determine the conditional thickness of

Component c
The conditional thickness of offered is

The conditional circulation purpose of the offered is

Make use of the limited and conditional circulation features to confirm the tend to be separate or otherwise not.

Step of 6

d)
Look at the tend to be separate activities or otherwise not.
The limited circulation of is:
The conditional thickness purpose of is
If tend to be separate activities after that proceed with the under guideline,

Thus, the tend to be separate activities.

Component d
The tend to be separate activities.

It’s seen that both limited and conditional circulation features provides the exact same thickness purpose therefore, predicated on this research to express being separate activities.

Make use of the limited circulation purpose of the to determine the worthiness of

Action 5 of 6

age)
Determine the worthiness of

Component age
The likelihood worth is 0.1548.

The likelihood worth of the arbitrary variable is lower than 0.5 and random adjustable is lower than 0.5 is 0.1548.

Make use of the limited circulation purpose of the to determine the worthiness of

Action 6 of 6

f)
Determine the worthiness of

Component age
The likelihood worth is 0.000.

The likelihood worth of is 0.000. Because, the constant circulation purpose assumption the likelihood worth at any certain point is zero.

Solution

Component a
The limited circulation purpose of is

Component b
The limited circulation purpose of is

Component c
The conditional thickness of offered is

Component d
The tend to be separate activities.

Component age
The likelihood worth is 0.1548.

Component age
The likelihood worth is 0.000.

Solution just
Component a
The limited circulation purpose of is

Component b
The limited circulation purpose of is

Component c
The conditional thickness of offered is

Component d
The tend to be separate activities.

Component age
The likelihood worth is 0.1548.

Component age
The likelihood worth is 0.000.

X and ,Y
Sn (x, у) XY
for several x, y
Roentgen
P(хY) € roentgen (ху) dxdy roentgen
Х.
fxY(y)e) x>0 and y> 0 usually
Х.
(x, y)dy (x)=f -fe(dy 0 dy =e*
X
x (х) —е’ х
Y.
fxY(y)e) x>0 and y> 0 usually
Y.
(y)(x,y)de (a*)dx *dy = age 0 =e
Y
S(y) age y
X offered Y
X offered Y
f(xIY=y) (x,y) (y) (x+y) age e*e age = age
X
Y
f(X IY y) age
X and Y
X and Y
X
x (х) —е’ х
(X|Y y)
X and Y
f(X IY y) fr(x) = age
X and Y
(age)
X and Y
X and Y
P(X
Р(x
P(X
0.5 (0.3934693) (0.3934693) = 0.154818 (Round to 4 decimal destination) = 0.1548
X
Y
X and Y
Р(х %3D0.5,Y
Р(х %3D0.5,Y
Р(х – 0.5, Y
(X 0.5,Y
X
x (х) —е’ х
X
x (х) —е’ х
X
Jx (x) -е”
X
Sx (х) —е” -ху
X
fx(x)e
Y
S(y) age y
Y
S(y) age y
Y
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