# Joint cumulative distribution function

1. Dec 9, 2015

### Linder88

1. The problem statement, all variables and given/known data
Compute the joint cumulative distribution function $F_XY(x,y)$?
2. Relevant equations
The marginal distribution function $F_X(x)$

F_X(x)=P(X\leq x)=
\begin{cases}
0,x<0\\
0.6,0\leq x<1\\
1,x\geq 1
\end{cases}

and $F_Y(y)$

F_Y=
\begin{cases}
0,y<0\\
0.3,0\leq y<1\\
0.7,1\leq y <2\\
1,y\geq 2
\end{cases}

3. The attempt at a solution
For independent (I know the are not) random variables X and Y

F_XY(x,y)=F_X(x)F_Y(y)=\\
[0.6u(x)+0.4u(x-1)][0.3u(y)+0.4u(y-1)+0.3u(y-2)]=\\
0.6*0.3u(x)u(y)+0.6*0.4u(x)u(y-1)+0.6*0.3u(x)u(y-2)+0.4*0.3u(x-1)u(y)+0.4*0.4u(x-1)u(y-1)0.4*0.3u(x-1)u(y-2)=\\
0.18u(x)u(y)+0.24u(x)u(y-1)+0.18u(x)u(y-2)+0.12u(x-1)u(y)+0.16u(x-1)u(y-1)+0.12u(x-1)u(y-2)

2. Dec 9, 2015

### ElijahRockers

Why are these variables not independent?

If they aren't, then Fxy is not equal to FxFy

3. Dec 10, 2015

### Linder88

My teacher told they are not independent even though I wish they were

4. Dec 10, 2015

### Ray Vickson

If all you are told are the two marginals, then it is impossible to give the joint distribution. Are you not told anything else at all about the two random variables?

5. Dec 10, 2015

### WWGD

6. Dec 10, 2015

### Linder88

Well, the whole question reads like in the attached picture but I already did the first part!

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7. Dec 10, 2015

### Ray Vickson

Just apply the DEFINITION of the joint cdf $F_{XY}(x,y)$. You will be able to present the results in a $2 \times 3$ table of $F(x,y)$ values, corresponding to $x = 0,1$ and $y = 0,1,2$.

8. Dec 10, 2015

### Linder88

I guess you mean

F_{XY}(x,y)=
\begin{cases}
(0.2+0.3+0.1)(0.2+0.1),x=0;y=0\\
(0.2+0.1+0.1)(0.3+0.1),x=1;y=1\\
0.2+0.1,y=2
\end{cases}

9. Dec 10, 2015

### Ray Vickson

No, I do not mean that. For one thing, it is completely wrong.

Let me repeat my previous question: what is the DEFINITION of $F_{XY}(x,y)$?

Expanded question: for a given pair $(x,y)$, how would you compute that?

10. Dec 14, 2015

### Linder88

I think i finally get it. For a given pair i would have that
$$F_{XY}(x,y)= \begin{cases} 0,x<0,y<0\\ 0.2+0.1,0\leq x<1,0\leq y<1\\ 0.2+0.1+0.3,0\leq x<1,1\leq y<2\\ 0.2+0.1+0.3+0.1,1\leq x<2,1\leq y<2\\ 0.2+0.1+0.3+0.1+0.2+0.1,1\leq x,2\leq y \end{cases}$$
or
$$F_{XY}(x,y)= \begin{cases} 0,x<0,y<0\\ 0.3,0\leq x<1,0\leq y<1\\ 0.6,0\leq x<1,1\leq y<2\\ 0.7,1\leq x<2,1\leq y<2\\ 1,1\leq x,2\leq y \end{cases}$$
Thanks.