Question

Let X be a random variable. Given the function f(x) = k(2x + 1),

for x = 0,1,2,3,4, find k such that f(x) is a probability mass function for X.

Answer 1

To find k such that f(x) = k(2x + 1) is a PMF for X, we calculate the sum of all probabilities for x values 0 to 4 and set it equal to 1. The sum is k multiplied by the series 1 + 3 + 5 + 7 + 9, which equals 25k. Hence, k = 1/25.

Step-by-step explanation:

To find the value of k such that the function f(x) = k(2x + 1) is a probability mass function (PMF) for the random variable X, we must first understand that a PMF must satisfy two main properties:

• The probability P(x) for any value of x must be between 0 and 1, inclusively.
• The sum of all probabilities for all possible values of x must equal to 1.

In this case, the random variable X takes on the values 0, 1, 2, 3, 4. To find k, we apply the second property of PMFs:

Σ P(x) = k(2×0 + 1) + k(2×1 + 1) + k(2×2 + 1) + k(2×3 + 1) + k(2×4 + 1) = k(1 + 3 + 5 + 7 + 9) = k×25 = 1

By solving this equation for k, we find that k = 1/25, which makes the function a legitimate PMF for X.