Question

example: identifying probability distributions
determine whether each distribution is a probability distribution. explain your reasoning.
2

Explanation:

Step1: Check non - negativity of probabilities

A probability distribution must have \(P(x)\geq0\) for all \(x\). Here, when \(x = 4\), \(P(4)=- 1\lt0\).

Step2: Check sum of probabilities

The sum of probabilities in a probability distribution should be \(1\). Let's calculate \(\sum P(x)=\frac{1}{2}+\frac{1}{4}+\frac{5}{4}+(-1)\). First, find a common denominator (4): \(\frac{2 + 1+5 - 4}{4}=\frac{4}{4} = 1\). But since \(P(4)=-1\) (violates non - negativity), the distribution is not valid.

Answer:

This is not a probability distribution because \(P(4)=-1\lt0\) (violates the non - negativity rule of probability distributions).