A joint table

Topic: Joint distributions · more questions on this topic
‹ Minimum and average of uniform draws A probability as an integral ›

The joint probability mass function of X and Y is

x\backslash y 1 3 6 8
2 0 0.1 0.1 0
5 0.2 0 0 0
6 0 0.2 0.1 0.3

  1. Find the marginal mass functions of X and of Y.

  2. Are X and Y independent?

  3. Calculate \mathbb{E}(X^2 Y).

  1. Row sums: \mathbb{P}(X=2)=0.2, \mathbb{P}(X=5)=0.2, \mathbb{P}(X=6)=0.6. Column sums: \mathbb{P}(Y=1)=0.2, \mathbb{P}(Y=3)=0.3, \mathbb{P}(Y=6)=0.2, \mathbb{P}(Y=8)=0.3.

  2. They are not independent. For example \mathbb{P}(X=2,Y=1)=0, while \mathbb{P}(X=2)\mathbb{P}(Y=1)=0.04.

  3. Only the six positive cells contribute:

\begin{aligned} \mathbb{E}(X^2 Y) &=0.1\cdot 4\cdot 3+0.1\cdot 4\cdot 6+0.2\cdot 25\cdot 1\\ &\quad+0.2\cdot 36\cdot 3+0.1\cdot 36\cdot 6+0.3\cdot 36\cdot 8\\ &=138.2. \end{aligned}