## Related questions with answers

Determine whether the two random variables, given in the following way, are dependent or independent. Let X denote the number of times a certain numerical control machine will malfunction: 1, 2, or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as

$\begin{matrix} & & & x & \\ \hline & & 1 & 2 & 3 \\ & 1 & 0.05 & 0.05 & 0.10 \\ y & 3 & 0.05 & 0.10 & 0.35 \\ & 5 & 0.00 & 0.20 & 0.10 \end{matrix}$

Solution

VerifiedThe variables $X$ and $Y$ will be $\text{\underline{independent}}$ if the equality

$\begin{align} f(x,y)=g(x)h(y) \end{align}$

holds for $\textbf{all}$ values $x$ and $y$ which the variables are able to assume.

In Exercise 3.49, we found the values

$g(1)=0.1 \text{ and } h(5)=0.3$

and from the joint distribution table we obtain

$f(1,5)=0$

But obviously,

$\underline{ f(1,5) }= 0 \underline{ \neq } 0.03 = 0.1 \cdot 0.3 =\underline{ g(1)h(5)}$

so the identity $(1)$ isn't true for all values of $x$ and $y$ and therefore the random variables $X$ and $Y$ $\textbf{aren't independent}$.

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