The negative binomial distribution is sometimes defined in a different way as the distribution of the number of failures that precede the kth success. If the kth success occurs on the xth trial, it must be preceded by x - k failures. Thus, find the distribution of Y = X - k, where X has the distribution of Definition 4.
DEFINITION 4
NEGATIVE BINOMIAL DISTRIBUTION A random variable X has a negative binomial distribution and it is referred to as a negative binomial random variable if and only if
![](https://test.transtutors.com/qimg/c9647ea1-9f56-423a-86e9-7e5a75559fca.png)