xy x f x y x y f x y) is a valid joint pmf/pdf. jzflniddy
Transcript of xy x f x y x y f x y) is a valid joint pmf/pdf. jzflniddy
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, >×
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