You repeatedly draw a number at random from the numbers $1,2,…,10$.
Let $X$ be the number of drawings until the number $1$ appears and $Y$
be the number of drawings until the number $10$ appears. What is the
joint probability mass function of $X$ and $Y$? What are the
probability mass functions of $\max (X,Y)$ and $\min (X,Y)$?
Attempt
$X$ and $Y$ are geometric both with parameter $\dfrac{1}{10}$ since each number is equally likely to occur when picking a number at random. We need to find $P(X=x, Y=y)$. Here I get confused as to whether make the assumption that both $X$ are $Y$ are independent. Can we assume this? IF so , then
$$ P(X=x,Y=y) = (0.9)^{x-1} 0.1 (0.9)^{y-1} 0.1 $$
Now, for the next part notice that $max(X,Y)=n$ iff $X \leq n \; \text{or} Y \leq n$.hence,
$$ P(\max(Y,X) = n) = P(X \leq n) + P(Y \leq n) = \sum_{x=1}^n (0.9)^{x-1} 0.1 + \sum_{y=1}^n (0.9)^{y-1} 0.1 $$
Similarly,
$$ P( \min (Y,X) = m) = O(Y \geq n) + (X \geq n) = \sum_{y=n}^{\infty} (0.9)^{y-1} 0.1 + \sum_{ x =n}^{\infty} (0.9)^{x-1} 0.1 $$
Am I correct?
Best Answer
$X$ and $Y$ are not independent since if $P(X=1,Y=1) =0$ but $P(X=1)>0$ and $P(Y>1)>0$.
if $1\le x < y$,
$$Pr(X=x, Y=y) = (0.8)^{x-1}(0.1)(0.9)^{y-x-1}(0.1)=0.8^{x-1}(0.9)^{y-x-1}(0.1)^2$$
That is the first $x-1$ times can be anything besides the two numbers, followed by $1$, then the following $y-x-1$ times we can get any number besides $10$.
Try to work out what happens when $x=y$ and for the case where $x>y$.
Notice that we can't conclude $\max(X,Y)=n \iff X \le n\lor Y \le n$ as if $X=n-2$ and $Y=n-1$, then we ahve $X \le n \lor Y \le n$ being true but $\max(X,Y)=n-1$.
\begin{align} Pr(\max(X,Y)=n) &= Pr(Y > X \land Y=n) + Pr(X > Y \land X=n) \\ &= \sum_{x=1}^{n-1}Pr(X=x,Y=n) + \sum_{y=1}^{n-1}Pr(X=n,Y=y) \end{align}
and
\begin{align} Pr(\min(X,Y)=n) &= Pr(Y > X \land X=n) + Pr(X > Y \land Y=n) \\ &= \sum_{y=n+1}^{\infty}Pr(X=n,Y=y) + \sum_{y=n+1}^{\infty}Pr(X=x,Y=n) \end{align}