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Chapter 5: Problem 97

What are the two basic properties of every probability distribution?

Short Answer

Expert verified
The two properties of any probability distribution are: 1) All probabilities must be nonnegative, meaning they must be greater than or equal to 0 and less than or equal to 1 (\( 0 \leq P(x) \leq 1 \)). 2) The sum of all possible outcomes must equal to 1 (\( \sum P(x) = 1 \)).

Step by step solution

01

Define Probability Distribution

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. It's used in various disciplines like finance, insurance, and physics to assess risk, predict outcomes, and analyze trends.
02

Rule 1: Probabilities are nonnegative

The first fundamental property of a probability distribution is that each probability value, denoted as \( P(x) \), is between 0 and 1, inclusive. This is represented mathematically as \( 0 \leq P(x) \leq 1 \). No probability can be less than 0 or greater than 1. In terms of a real-life example, no event can occur less than 0 times, nor can it occur more than 100% of the time.
03

Rule 2: Sum of all probabilities equal one

The second fundamental property is that when you add up all the probabilities, the total should be equal to 1. This means the sum of probabilities of all possible outcomes must be 1. This is represented mathematically as \( \sum P(x) = 1 \). This reflects the principle that some outcome must occur, so the probabilities of all possible outcomes must add up to 100% or 1.

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Key Concepts

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

Probability Rules
In probability theory, a probability distribution adheres to specific rules that dictate how probabilities are assigned to events. These rules help ensure that every probability model is logical and consistent.

One of the most important elements is that the probability of any event is a numerical value between 0 and 1.
This reflects how likely an event is to occur, with 0 representing an impossible event and 1 a certainty.
  • A probability of 0 means the event will never happen. For example, the probability of rolling a 7 on a standard 6-sided die is 0.
  • A probability of 1 indicates certain occurrence. Rolling a number between 1 and 6 on a 6-sided die has a probability of 1 because one of those numbers is guaranteed to show up.
Understanding probability rules helps in evaluating models and theories in real-world situations, ensuring predictions are realistic and accurate.
Nonnegative Probabilities
Nonnegative probabilities are a cornerstone of probability distributions.
Simply put, they refer to probabilities that cannot be less than 0.
In practical terms, this means you can't have a negative chance of something happening.
These probabilities must always be between 0 and 1, where fractions or decimal forms of percentages are used to express likelihood.

This property is crucial because it aligns with our intuitive understanding of probability.
Seeing as probabilities express possible outcomes occurring, it would be nonsensical to represent an event occurring with a negative number.
For example:
  • The phrase "There's a -25% chance of rain" doesn't make logical sense, just as there couldn't be a negative quantity of occurrences in our understanding of reality.
Rather, probabilities depict the scale of possibility from nonexistent (0) to certainty (1).
Sum of Probabilities Equals One
The principle that the sum of probabilities equals one is another fundamental aspect of probability distributions.
When dealing with multiple possible outcomes of a random event, the probabilities of all individual outcomes combined must result to 1.

This total reflects the certitude that one of these possible outcomes must occur.
Mathematically, this is shown as: \[\sum P(x) = 1\]For example:
  • If you flip a coin, there are two possible outcomes: heads or tails. If both have an equally likely chance of occurring, each has a probability of 0.5. Added together, these probabilities sum to 1, affirming that one outcome will indeed occur.
This rule ensures a complete account of all possibilities, confirming that the distribution is valid and reflective of real scenarios.

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Most popular questions from this chapter

A box contains 10 items, of which 3 are defective and 7 are nondefective. Two items are randomly selected, one at a time, with replacement, and \(x\) is the number of defectives in the sample of two. Explain why \(x\) is a binomial random variable.

"How many TVs are there in your household?" was one of the questions on a questionnaire sent to 5000 people in Japan. The collected data resulted in the following distribution: $$\begin{array}{l|cccccc}\hline \text { Number of TVs/Household } & 0 & 1 & 2 & 3 & 4 & 5 \text { or more } \\\\\text { Percentage } & 1.9 & 31.4 & 23.0 & 24.4 & 13.0 & 6.3 \\\\\hline\end{array}$$ a. What percentage of the households have at least one television? b. What percentage of the households have at most three televisions? c. What percentage of the households have three or more televisions? d. Is this a binomial probability experiment? Justify your answer. e. Let \(x\) be the number of televisions per household. Is this a probability distribution? Explain. f. Assign \(x=5\) for "5 or more" and find the mean and standard deviation of \(x.\)

Census data are often used to obtain probability distributions for various random variables. Census data for families in a particular state with a combined income of 50,000 dollar or more show that \(20 \%\) of these families have no children, \(30 \%\) have one child, \(40 \%\) have two children, and \(10 \%\) have three children. From this information, construct the probability distribution for \(x,\) where \(x\) represents the number of children per family for this income group.

Express the tossing of one coin as a probability distribution of \(x,\) the number of heads occurring (that is, \(x=1\) if a head occurs and \(x=0\) if a tail occurs).

Find the mean and standard deviation of \(x\) for each of the following binomial random variables: a. The number of tails seen in 50 tosses of a quarter b. The number of left-handed students in a classroom of 40 students (Assume that \(11 \%\) of the population is left-handed.) c. The number of cars found to have unsafe tires the 400 cars stopped at a roadblock for inspection (Assume that \(6 \%\) of all cars have one or more unsafe tires.) d. The number of melon seeds that germinate when a package of 50 seeds is planted (The package states that the probability of germination is 0.88.)
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