Let $X$ be the number that shows up when rolling a die. Now throw another dice $X$ times $(Y_1, ..., Y_X)$ and calculate the sum $Z = \sum_{k=1}^X Y_k$.
What kind of Stochastic Process is this? How do you calculate mean value and variance of $Z$?
Let $X$ be the number that shows up when rolling a die. Now throw another dice $X$ times $(Y_1, ..., Y_X)$ and calculate the sum $Z = \sum_{k=1}^X Y_k$.
What kind of Stochastic Process is this? How do you calculate mean value and variance of $Z$?
Look up Wald's Lemma. This lemma is applicable here as $X$ and $Y_k$ are independent. Variance also can be found using this lemma and using the fact that $Z^2=\sum_{k=1}^{X}Y_k^2+\sum_{i
It is a compound probability distribution.
You can marginalize by taking the average of the six pmfs, giving a distribution that looks like $p(1) = 1/6^2,\dots, p(36) = 1/6^7$ and then you can find the variance of this finite marginal distribution directly. For the mean you can just take the mean of the six distribution means.
Your random variable is also a branching process $(Z_n)$ observed at time $n=2$, where the offspring distribution is given by a single roll of the die. You want to substitute $n=2$ below, where $\mu$ and $\sigma^2$ are the mean and variance for a single die roll $\mathbb{E}(Z_n)=\mu^n\quad\mbox{ and }\quad\mathbb{V}(Z_n)={\sigma^2\mu^n(\mu^n-1)\over\mu^2-\mu}.$