0

I have a random vector $(X,Y,Z,W)$ with the following Probability Mass Function:

enter image description here

The joint probability marginal distribution of $(X,Y)$ in a similar table with its own marginals is:

$$(0,0)=0.225$$ $$(1,0)=0.2$$ $$(0,1)=0.275$$ $$(1,1)=0.3$$

  • How can I find the joint probability distribution function $F_{(x,y)}(x,y)$

  • How can I calculate probabilities such as $P\{X+Y \geq Z+W\}$, $P\{1 \geq X+Y |Z+W \geq 2\}$ ?

Gotey
  • 253

1 Answers1

1

The joint PMF of $(X,Y)$ is given by $\sum_{z,w} p(x,y,z,w)$.

Then, $F_{(X,Y)}(x,y) = \sum_{t \leq x, u \leq y} p(t,u)$.

From the definition, $P(X+Y \geq Z+W) = \sum_{x+y \leq z+w} p(x,y,z,w)$ (write out the quadruples of $(x,y,z,w)$ that satisfy this inequality and then sum the pmf over them).

Similarly, from teh definition, To calculate $P(1 \geq X+Y | Z+W \geq 2) = \frac{P(1 \geq X+Y \text{ and } Z+W \geq 2)}{P(Z+W \geq 2)} = \frac{ \sum_{1 \geq x+y \text{ and } z+w \geq 2} p(x,y,z,w)}{\sum_{x,y, z+w \geq2} p(x,y,z,w)}$.

Batman
  • 19,390
  • Hi, thanks for your reply. First off i would like to clarify some concepts i don't understand: what would be $u$ and $t$?, How it's called the notation you put under the summation signs and what exactly means? – Gotey Dec 26 '15 at 15:38
  • 1
    Summing over those variables. $\sum_{z,w}$ means sum over all values of z,w. $\sum_{t \leq x, u \leq y}$ means sum over all values $t,u$ which are less than or equal to $x,y$ respectively. $\sum_{x,y,z+w \geq 2}$ means sum over all values of x, all values of y, and all pairs (z,w) such that $z+w \geq 2$. – Batman Dec 26 '15 at 15:51