Question

Which of the following could be a valid pmf (probability mass function) of a random variable x?

a. p(x=k) = 2k - 1 / 2n - 1, for k ∈ {1, 2, …, n}
b. p(x=k) = 1/k, for k ∈ {2, 3, 4, …}
c. p(x=k) = 1/k(k - 1), for k ∈ {1, 2, …, n}
d. p(x=k) = 1/2^k, for k ∈ {1, 2, …, n}

Answer 1

Option a and d could be valid pmfs of a random variable x.

Step-by-step explanation:

A pmf (probability mass function) represents the probabilities of each possible value of a random variable. Let's evaluate each option:

1. a. p(x=k) = (2k - 1) / (2n - 1) - This is a valid pmf as it satisfies two conditions: each probability value is between zero and one, and the sum of the probabilities equals one.
2. b. p(x=k) = 1/k - This is not a valid pmf because the sum of the probabilities does not equal one for the given values of k. The sum should be infinite.
3. c. p(x=k) = 1/(k(k - 1)) - This is not a valid pmf because some probabilities are negative and the sum of the probabilities for the given values of k is not equal to one.
4. d. p(x=k) = 1/2^k - This is a valid pmf as each probability value is between zero and one, and the sum of the probabilities equals one.

In conclusion, options a and d could be valid pmfs of a random variable x.