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We have 2 bags, each one contains two white balls and one black. We take 2 balls from each bag. Let $X_1$ - number of white balls that we got from first bag, $X_2$ - from second. Find CDF of variable $Z=\max(X_1,X_2)$.

$F_Z\left(t\right)=\mathbb{P}\left(Z\le t \right)=\mathbb{P}\left(\max\left(X_1,X_2 \right)\le t \right)=?$

And what next?

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  1. Write out the support of the random variables $X_1$ and $X_2$. What values can these random variables admit?
  2. What are the respective probabilities of observing each outcome in the support?
  3. Are $X_1$ and $X_2$ independent random variables? Why or why not?
  4. What are the probabilities of observing the joint outcomes $(X_1, X_2) = (x_1, x_2)$?
  5. For each joint outcome $(x_1, x_2)$, what is the value of $z = \max(x_1, x_2)$ and what is the probability $\Pr[Z = z]$?
  6. What is the cumulative probability $\Pr[Z \le z]$ based on your calculation of the probability mass function in step (5)?
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    1. We can get 0,1 or 2 balls 2. $\mathbb{P}(X_i = 0)=0$, $\mathbb{P}(X_i = 1)=3\cdot\frac{1}{3}\cdot\frac{2}{3}$, $\mathbb{P}(X_i = 2)=3\cdot\frac{2}{3}\cdot\frac{2}{3}$ 3. Yes, they are independent since getting ball from 1 bag doesn't affect another bag 4. If $X_1$ and $X_2$ are independent, then $\mathbb{P}(X_1=m)\cdot\mathbb{P}(X_2=n)=\mathbb{P}(X_1=m \text{ and } X_2=n)$ 5. $z=x_1$ for $x_1>x_2$ and $z=x_2$ for $x2>x_1$ and $z=x_1=x_2$ for $x_1=x_2$ – 2017-02-07
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    As for the rest, what is the probability $Pr[Z=z]$, hmm, I am really not sure. Maybe $Pr[Z=z]=Pr(Z=\max(x_1,x_2))=Pr\left(Z=x_1 \text{ and } x_1>x_2\right) + Pr\left(Z=x_2 \text{ and } x_2 > x_1 \right) + Pr\left(Z=x_i \text{ and } x_1=x_2 \right) $? – 2017-02-07