Example

Suppose that the joint pmf/pdf of X ,Y is given by

f (x , y) =xy x�1

3for x = 1, 2, 3 0 < y < 1

Verify that f (x , y) is a valid joint pmf/pdf.

§ {tofkiy )dy =L [jzflniddy = 1

Example

Calculate Pr(Y � 1/2 and X � 2).

Ply >± ,×>z)

fln .y)=§xyM a -1,2 ,} 0<Y< I

P( yx 's,x%z)=PfAnB)a- I

13=1×32 ]=[X=2]U[x=3]

Past ,xm)=P( yttzntytplyzt ,x⇒)= fifty 'dyt Sjztssjidy =

÷fiYdy+ Sjdiay

=÷ Eli.

- ' HIJ,

....

Joint bivariate cdf

For ANY two random variables X ,Y we can characterize their jointdistribution using the joint cumulative distribution function F (x , y).

F (x , y) = Pr(X x and Y y)

If (X ,Y ) have a joint pdf f (x , y) then

F (x , y) =

Zx

�1

Zy

�1f (r , s)dr ds

and

f (x , y) =d2F (x , y)

dxdy

1- dim can fH=#dYT

Example

Suppose that the joint cdf of X ,Y is

F (x , y) =1

16xy(x + y) 0 x 2 0 y 2

Find the cdf of X (alone).

Ynoey's

:*y xx1

O2 -

:[YH ' o×!5±z

< Flu ,y ) F( 2 ,Y )

either ¢ k z

×< o O>

ory< o

0

Fdn ) = cdf of X alone = marginal caf of X

= Pr ( X en )= ykjma Fln , y) = ( from the picture)

= families:÷x " = fngnilin;°an "

1 if x >2

Example

Find the pdf of (X ,Y ).

Flmy )= ,TKy( nty ) Oenez OEYEZ

ftp.adf#yb=atyCte2xy+teyY=tEGx+2y]

=f(x+y) OEx±2 Oeyez

f( my )={$KtHit oerez , oeyez

0 otherwise

3.5 Marginal distributions

I Assume that X and Y are random variables having a jointdistribution (X ,Y ).

I The distribution of X (alone) is the marginal distribution of X .

I The distribution of Y (alone) is the marginal distribution of Y .

If (X ,Y ) have a discrete joint distribution with pmf f (x , y) the marginaldistribution of one variable can be calculated by summing f (x , y) over allpossible values of the other variable

f1(x) =X

All y

f (x , y)

f2(y) =X

All x

f (x , y)

Example

Recall the previous example

Find the marginal distributions of X and Y .

±:: 2

Y : 0.4 0.2 0.2 0.2

flat : fli ) fly fkd

ftp.yEfll ,Y)=f( 1,1 )tf( lift fell,

} )+f 11,4 )

Marginal distributions

If (X ,Y ) have a joint pdf f (x , y), the marginal pdf of one variable isfound by integrating out the values of the other variable.

f1(x) =

Z 1

�1f (x , y) dy

f2(y) =

Z 1

�1f (x , y) dx

Example

Suppose that the joint pdf of (X ,Y ) is given by

f (x , y) =

(214 x

2y if x2 < y < 1

0 otherwise

Derive the marginal distributions of X and Y .

t.mn#*:tItHYg

t.ly =fj¥xYdy = ¥ ' i III. = 2¥ xi ( t - ¥ ) = st tax ')

f ,(a) = Is ( set st ) -1en ± 1

fdy ) = [fH . Ddr = frrtytfiydx = ¥ y of μ= ¥ y 't . ( y

' 'T yk) = Is y"

f.ly )= Z y% o±y < 1

Example

Suppose that the joint pmf/pdf of X ,Y is given by

f (x , y) =xy x�1

3for x = 1, 2, 3 0 < y < 1

Derive the marginal pmf of X and the marginal pdf of Y .

f ,In ) = [ flmy )dy = fits nyt '

dy = § y"

lot = to

fill )=PrlX=D= I fH=Pr(x=z)= } Pr(X=3)=§

fdy )=¥aHnD = t + Fy + y'

O < y < 1

Independent random variables

Two random variables X ,Y are independent if their joint cdf F (x , y) canbe written as

F (x , y) = F1(x)F2(y)

F1(x) = Pr(X x) is the marginal cdf of XF2(y) = Pr(Y y) is the marginal cdf of Y .

Pr(X x ,Y y) = Pr(X x)Pr(Y y)

If X ,Y are independent and A,B are subintervals of the real line,

Pr(X 2 A,Y 2 B) = Pr(X 2 A)Pr(Y 2 B)

Independent random variables

If X ,Y are independent, then

I X ,Y are discrete, then, the joint pmf f (x , y) = Pr(X = x ,Y = y)factorizes into the product of the marginal pmfs

f (x , y) = f1(x)f2(y)

f1(x) = Pr(X = x) f2(y) = Pr(Y = y)

I X ,Y are continuous, then, the joint pdf factorizes into the productof the marginal pdfs

f (x , y) = f1(x)f2(y)

f1(x) is the marginal pdf of Xf2(y) is the marginal pdf of Y .

Example

Suppose that two measurements X and Y are made of the rainfall at acertain location on May 1 in two consecutive years. It might bereasonable, given knowledge of the history of rainfall on May 1, to treatthe random variables X and Y as independent. Suppose that the pdf gof each measurement is

g(x) =

(2x if 0 x 1

0 otherwise

Determine the value of Pr(X + Y 1).

flniytjointpdf

= ffhapdnay=GKt9ly)

hy

Tirana'

= })o4nydndY

then exercise

i, >×

XTYH