Suppose that in a sequence of independent bernoulli trails, the number of failures up to the first success is counted.

FIND: What is the frequency function for this random variable?

attempted SOLUTION:

We know that Bernoulli distribution where $f(k)=p^k(1-p)^{1-k}$ is the frequency function for number of successes in a single trial (??).

We also know that the geometric dirtribution models the number of failures up to the first success. Wouldnt be the frequency function for the random variable just be the geometric distribution with frequency function $f(k)=(1-p)^{k-1}p$ ?

provided SOLUTION

Our professor provided a solution to this exercises that states: $f(k)=(1-p)^{k}p$

Is my approach wrong?

  • $\begingroup$ The distribution you provide, $f(k) = (1-p)^{k-1}p$, gives the Frequency function for the first success being in $k$-th run. This corresponds to $k-1$ failures, not $k$ failures, which you seek. $\endgroup$ – denklo Jul 31 '18 at 13:17
  • $\begingroup$ FYI, the name "Frequency function" does not exist. $\endgroup$ – Did Jul 31 '18 at 13:18

If you choose k=1, the probability is (1-p)*p for (failure, success), so f(1)=(1-p)*p. So your proff. is right.

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