Final answer:
The random variable X follows a hypergeometric distribution, with its pmf given by the formula involving combinations of the total number of students, the number of nuclear engineering students, and the size of the sample. The support of X is {0, 1, 2, 3}.
Step-by-step explanation:
In the scenario provided, the random variable X, which represents the number of nuclear engineering students selected in a sample of 6 from a group of 16 students, follows a hypergeometric distribution. This is because the selections are made without replacement from a finite population that consists of two categories (nuclear engineering students and other students).
The probability mass function (pmf) of a hypergeometric distribution is given by:
P(X = k) = \(\frac{{\binom{K}{k} \cdot \binom{N - K}{n - k}}}{\binom{N}{n}}\)
where:
- N is the total number of students (16 in this case).
- K is the number of nuclear engineering students (3 in this case).
- n is the number of students in the sample (6 in this case).
- k is the number of nuclear engineering students in the sample.
The support of X is the set of possible values that k can take on, which in this scenario is {0, 1, 2, 3}, assuming that there are at least 6 non-nuclear engineering students in the overall group.