Roll a die repeatedly and let $X=\inf\{n:n\text{th roll is a six}\}$ and $Y=\inf\{n>X:n\text{th roll is a one}\}-X$, find the joint PMF

Question

We are repeatedly rolling a fair die. Let $X$ be the number of rolls needed to see the first $6.$ Let Y be the number of rolls needed to get the first $1$ after seeing the first $6.$ So $X=\inf\{n:n\text{th roll is a six}\}$ and $Y=\inf\{n>X:n\text{th roll is a one}\}-X.$

a) What is the distribution of $X?

b) Find the joint PMF of $X,Y$ and show that they are independent.

c) What is the distribution of $Y$?

$\textbf{My work so far:}$

For part (a), denote the probability distribution of $X$ by $\mu_X.$ Then for all $n\in\mathbb{N}$ I have $$\mu_X(n)=\mathbb{P}(X=n)=\frac{5^{n-1}}{6^n}.$$

For part (b), let $\textbf{n}=(n_1,n_2)$ where $n_1,n_2\in\mathbb{N},$ and let $\textbf{Z}=(X,Y).$ Then the probability mass function of $\textbf{Z}$ is given by $$p_{Z}(\textbf n)\begin{cases}0&\text{if }n_2\leq n_1\\\displaystyle\frac{5^{n_1-1}}{6^{n_1}}\cdot\frac{5^{n_2-n_1-1}}{6^{n_2-n_1}}&\text{if }n_1\leq n_2.\end{cases}$$ And it follows from this, if correct, that the two random variables are independent.

For part (c), I have that $$\mu_Y(n)=\frac{5^n}{6^n}.$$

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Is my work above accurate?

Any comments are welcomed and much appreciated.

Thank you for your time.

Answer 1

a) You're correct. It's actually Geometric distribution with $p=1/6$.

b) You're correct.

c) You missed $-1$ in power in numerator. Actually $Y$ has same distribution as $X$ (I assume that's a typo as you correctly wrote joint distribution).