Question

1. Let X the number of tires on a randomly selected automobile that are un- derinflated. a). Which of the following three functions, p, f, and g, is a legitimate p.m.f for X, and why are the other two not allowed? p(x) 0.3 0.2 0.1 0.05 0.05 f(x) 0.4 0.1 0.1 0.1 0.3 g(x) 0.4 0.1 0.2 0.1 0.3

Answer 1

We know that sum of PMf over all possible values is equal to 1. So

We know that sum of PMf over all possible values is equal to 1. So Sigma P_X (x) = 1 In the first case, we have Sigma P_X (x) = (P_X (0) + P_X (1) + ... P_X (4)) = 0.3 + 0.2 + 0.1 + 0.05 = 0.7 So this is less than 1 and thus can't be a pmf. Second, we have Sigma P_X (x) = (P_X (0) + P_X (1) + ... P_X (4)) = 0.4 + 0.1 + 0.1 + 0.1 + 0.3 = 1 So this is equal to 1 and is a pmf. The third case is Sigma P_X (x) = (P_X (0) + P_X (1) + ... P_X (4)) = 0.4 + 0.1 + 0.2 + 0.1 + 0.3 = 1.1 This is greater than 1, thus it can't be a pmf.

In the first case, we have

So this is less than 1 and thus can't be a pmf.

Second, we have

So this is equal to 1 and is a pmf.

The third case is

This is greater than 1, thus it can't be a pmf.