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The failures-only convention counts only the number of failures before the r-th success, whereas the total-trials convention counts all trials. The failures-only variable is shifted by r compared to the total-trials variable.

Conditions: Comparing two different definitions of the negative binomial random variable.

The failures-only convention counts only the number of failures before the r-th success, whereas the total-trials convention counts all trials. The failures-only variable is shifted by r compared to the total-trials variable.

Conditions: Comparing two different definitions of the negative binomial random variable.

Under the total-trials convention, the mean number of trials needed to achieve r successes is $r/p$.

Conditions: Using the total-trials convention (counting all trials).; Success probability $p > 0$.

Under the total-trials convention, the mean number of trials needed to achieve r successes is $r/p$.

Conditions: Using the total-trials convention (counting all trials).; Success probability $p > 0$.

To finish on trial x, the first $x-1$ trials must contain exactly $r-1$ successes, and the x-th trial must be a success. The probability is calculated as the number of ways to arrange $r-1$ successes in $x-1$ trials, multiplied by the probability of that specific sequence.

Conditions: Trials are independent.; Success probability p is constant.; $x \ge r$.

To finish on trial x, the first $x-1$ trials must contain exactly $r-1$ successes, and the x-th trial must be a success. The probability is calculated as the number of ways to arrange $r-1$ successes in $x-1$ trials, multiplied by the probability of that specific sequence.

Conditions: Trials are independent.; Success probability p is constant.; $x \ge r$.