I'm looking up Geometric random variables, where $X_1, X_2....$ are independent identically distributed variables which are $Ber(p)$. The book says,

$$ Y = \min \{n\geq 1| X_n = 1\} \sim Geo(p)$$ However, it then goes on to say,

$ Y \in\mathbb N$ . This is where I'm confused. I guess I don't properly understand the previous line.


  • $\begingroup$ I would say that: We provide a further attempt, the success of which has a probability 'p'. Geo (p) is the probability that the first success occurs exactly in the n-th attempt (and n-1 preceding the attempts failed). For example $Y = 3 \Rightarrow $ the first two experiments negative and the third positive: $P(Y=3)=p\cdot(1-p)^3$ $\endgroup$ – georg Sep 3 '14 at 13:26

$Y$ is defined as the minimal $n$ for which $X_n$ is $1$ (or true). Since the $X_i$ are numbered with $i\in \mathbb N$, we have that $n\in \mathbb N$, and thus $Y\in \mathbb N$.


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