# For which values are the variables independent?

• MHB
• mathmari
In summary: Indeed. (Nod)Similarly we can skip $X=1$ after checking $X=-1$ and $X=0$.In summary, we can check if $X$ and $Y$ are independent by checking if $P[X=m,\ Y=n]=P[X=m]\cdot P[Y=n]$.
mathmari
Gold Member
MHB
Hey! :giggle:

We have the table of distribution of $X$, $Y$ and their joint random variable :

with $$(c,d)\in \left \{(c,d)\in \mathbb{R}^2\mid 0\leq c\leq \frac{1}{4}, \ 0\leq d\leq \frac{1}{2}, \ \frac{1}{4}\leq c+d\leq \frac{1}{2}\right \}$$

I want to calculate the values of $c$ and $d$ such that $X$ and $Y$ are independent.
So do we have to check each combination so that $P[X=m,\ Y=n]=P[X=m]\cdot P[Y=n]$ ?

\begin{align*}&P[X=-1, \ Y=0]=c \ \text{ and } \ P[X=-1]\cdot P[Y=0]=\frac{1}{4}\cdot \frac{1}{2}=\frac{1}{8} \text{ so } c=\frac{1}{8} \\ &P[X=0, \ Y=0]=d \ \text{ and } \ P[X=0]\cdot P[Y=0]=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4} \text{ so } d=\frac{1}{4}\end{align*}
We check now also the others if $X$ and $Y$ are indeed independent (or do we not have to? :unsure: )
\begin{align*}&P[X=-1, \ Y=1]=\frac{1}{4}-c=\frac{1}{4}-\frac{1}{8}=\frac{1}{8} \ \text{ and } \ P[X=-1]\cdot P[Y=1]=\frac{1}{4}\cdot \frac{1}{2}=\frac{1}{8} \text{ so correct} \\ &P[X=0, \ Y=0]=d=\frac{1}{4} \ \text{ and } \ P[X=0]\cdot P[Y=0]=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4} \text{ so correct} \\ &P[X=0, \ Y=1]=\frac{1}{2}-d=\frac{1}{2}-\frac{1}{4}=\frac{1}{4} \ \text{ and } \ P[X=0]\cdot P[Y=1]=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4} \text{ so correct}\\ &P[X=1, \ Y=0]=\frac{1}{2}-c-d=\frac{1}{2}-\frac{1}{8}-\frac{1}{4}=\frac{1}{8} \ \text{ and } \ P[X=1]\cdot P[Y=0]=\frac{1}{4}\cdot \frac{1}{2}=\frac{1}{8} \text{ so correct}\\ &P[X=1, \ Y=1]=c+d-\frac{1}{4}=\frac{1}{8}+\frac{1}{4}-\frac{1}{4}=\frac{1}{8} \ \text{ and } \ P[X=1]\cdot P[Y=1]=\frac{1}{4}\cdot \frac{1}{2}=\frac{1}{8} \text{ so correct} \end{align*}:unsure:

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Hey mathmari!

Seems correct to me. (Nod)

We might skip a couple of them, since we know that if $A$ and $B$ are independent, that $A$ and $B^c$ are also independent.

Klaas van Aarsen said:
We might skip a couple of them, since we know that if $A$ and $B$ are independent, that $A$ and $B^c$ are also independent.

You mean to check only with $Y=0$ and not with $Y=1$ ? :unsure:

mathmari said:
You mean to check only with $Y=0$ and not with $Y=1$ ?
Indeed. (Nod)

Similarly we can skip $X=1$ after checking $X=-1$ and $X=0$.
We can prove it if we want to.

## 1. What does it mean for variables to be independent?

When two variables are independent, it means that there is no relationship or correlation between them. This means that changes in one variable do not affect the other variable.

## 2. How can I determine if two variables are independent?

To determine if two variables are independent, you can conduct a statistical analysis such as a correlation or regression test. If the results show a low correlation coefficient or a p-value above the significance level, then the variables can be considered independent.

## 3. Can variables be partially independent?

No, variables can only be considered independent or dependent. Partial independence is not possible as it would imply that there is some level of correlation between the variables.

## 4. Are independent variables always the cause of dependent variables?

No, just because two variables are independent does not mean that one causes the other. It simply means that there is no relationship between them.

## 5. Can two variables be independent in one situation and dependent in another?

Yes, the relationship between variables can change depending on the context or situation. Two variables can be independent in one scenario and dependent in another, depending on the factors at play.

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