Question

Suppose

X,Y

are continuous random variables, with cumulative distribution functions (cdfs)

F

X

and

F

Y

, respectively. For each of the following, determine whether the function

F

is

necessarily the cdf of some random variable

Z

? In case the function is a cdf, find the density

f

Z

in terms of

F

X

,

f

X

,

F

Y

and

f

Y

. If the function is not necessarily a cdf, give an example

of random variables

X,Y

such that the function is not a cdf.

(a)

F

(

t

) =

F

X

(

t

)+2

F

Y

(

t

)

3

.

(b)

F

(

t

) =

F

X

(

t

)

·

F

Y

(

t

)

.

(c)

F

(

t

) =

F

X

(

F

Y

(

t

))

.

Answer 1

Answer

F(t) = Fx ( Fy(t) )

because the CDF of F(t) is in terms of Fx and Fy but Fy depends of t