After how many trials can I expect my first success?
Geometric
The geometric distribution models the number of Bernoulli trials needed for the first success. Its PMF is: [ P(X=k) = (1-p)^{k-1} p ] Where:
- ( p ) is the probability of success on each trial.
- Expected value (mean) = ( \frac{1}{p} )
- Variance = ( \frac{1-p}{p^2} )
Q4) How many trials are needed to achieve the r-th success?
Negative Binomial
The negative binomial distribution models the number of trials needed to achieve a fixed number of successes. Its PMF is: [ P(X=k) = \binom{k-1}{r-1} p^r (1-p)^{k-r} ] Where:
- ( r ) is the desired number of successes.
- ( p ) is the probability of success on each trial.
- Expected value (mean) = ( \frac{r}{p} )
- Variance = ( \frac{r(1-p)}{p^2} )