Hello i am relatively new to statistics and I struggle with a problem like this:
Assuming the chance of a fatal accident is 0.01 each day given that it hasn't yet happened. When can you expect it to happen, assuming that the fatal accident can only happen once?
As far as I can understand I cannot use Expected Value because the possible outcomes are infinite.
How can I tackle this problem?
The expected value can, in fact, be expressed as an infinite sum. If we let $T$ be the number of days until the accident occurs, and assume independence, we have \begin{align*} \mathbb P(T=1)&= 0.01\\ \mathbb P(T=2)&=(1-0.01)(0.01)\\ \mathbb P(T=3)&= (1-0.01)^2(0.01)\\ \mathbb P(T=t)&= (1-0.01)^{t-1}(0.01) \end{align*} In other words $T$ follows a geometric distribution with parameter $0.01$, and support $\{1,2,\cdots\}$. The expected value can be found with the following infinite sum: \begin{align*} \mathbb E(T)&= \sum_{t=1}^\infty t\,\mathbb P(T=t)=\sum_{t=1}^\infty t(0.99)^{t-1}(0.01)=\cdots =\frac{1}{0.01}=100 \end{align*}