Final answer:
The value of k in the given pmf is -1/3.
Step-by-step explanation:
To find the value of k in the given pmf, we need to use the properties of a probability mass function (pmf). The pmf for the given random variable X is given by p(x) = (x + 6k)/3 for x = 1,2,3 and p(x) = 0 for all other values of x. Since p(x) is a probability, it must satisfy two conditions:
1. Each p(x) must be between zero and one, inclusive.
2. The sum of all p(x) must be equal to one.
Using these conditions, we can calculate the value of k by setting up the equation:
(1 + 6k)/3 + (2 + 6k)/3 + (3 + 6k)/3 = 1
Simplifying this equation gives:
3 + 6k + 2 + 6k + 3 + 6k = 3
Combining like terms, we get:
15k + 8 = 3
Subtracting 8 from both sides:
15k = -5
Dividing by 15:
k = -1/3
Therefore, the value of k is -1/3.