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Chapter 6: Problem 3

What are the two requirements for a discrete probability distribution?

Short Answer

Expert verified
The two requirements are: the sum of all probabilities must equal 1, and each probability must be between 0 and 1, inclusive.

Step by step solution

01

Identify the Total Sum Requirement

A discrete probability distribution must satisfy the requirement that the sum of all probabilities is equal to 1. Mathematically, this can be written as: \[ \forall x_i \text{ in the sample space, } \text{sum} \, P(x_i) = 1 \] This ensures that we account for all possible outcomes.
02

Identify the Non-Negative Requirement

Each individual probability in a discrete probability distribution must be non-negative. That is, for any event \( x_i \), its probability \( P(x_i) \) must satisfy: \[ \forall x_i, \, 0 \leq P(x_i) \leq 1 \] This ensures that probabilities are meaningful, as they cannot exceed 1 or be less than 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Total Sum Requirement
In a discrete probability distribution, the total sum of all the individual probabilities must equal 1.
This concept is vital because it ensures that we account for every possible outcome within our sample space.
Mathematically, this is stated as: \[ \forall x_i \text{ in the sample space, } \text{sum} \, P(x_i) = 1 \]Here, \( x_i \) represents each possible outcome, and \( P(x_i) \) is the probability of that outcome.
When we add up all these probabilities, the result should be exactly 1.
This means we have considered every potential situation and nothing is left out. If the total is not 1, then something is wrong with our probability distribution.
Non-Negative Requirement
Each probability in a discrete probability distribution must be non-negative.
This ensures that probabilities are realistic and meaningful. Specifically, for any event \( x_i \), the probability \( P(x_i) \) must comply with:\[ \forall x_i, \, 0 \leq P(x_i) \leq 1 \]This formula means every individual probability, \( P(x_i) \), should be a number between 0 and 1 inclusive.
It cannot be negative, because negative probabilities don't make sense in real-life scenarios.
Additionally, it cannot exceed 1, because 1 represents absolute certainty that the event will occur. Keeping probabilities within this range helps maintain their validity and usefulness.
Probability Rules
Besides the total sum and non-negative requirements, there are other essential probability rules to keep in mind.
Here are some key rules:
  • The probability of an impossible event is 0. This means if an event cannot happen, its probability is zero.
  • The probability of a certain event is 1. This implies if an event is guaranteed to happen, its probability is one.
  • If two events, A and B, are mutually exclusive, the probability of either event occurring is the sum of their individual probabilities. This can be written as:
    \[ P(A \, or \, B) = P(A) + P(B) \]These rules help ensure that all probabilities are handled correctly and logically.
    They are foundational principles you should always consider when dealing with discrete probability distributions.

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