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Chapter 5: Problem 3

Consider each distribution. Determine if it is a valid probability distribution or not, and explain your answer. (a)$$\begin{array}{c|ccc} \hline x & 0 & 1 & 2 \\ \hline P(x) & 0.25 & 0.60 & 0.15 \\ \hline \end{array}$$ (b)$$\begin{array}{c|ccc} \hline x & 0 & 1 & 2 \\ \hline P(x) & 0.25 & 0.60 & 0.20 \\ \hline \end{array}$$

Short Answer

Expert verified
Distribution (a) is valid; distribution (b) is not valid.

Step by step solution

01

Understanding a Probability Distribution

A probability distribution assigns probabilities to each outcome of a random variable. For it to be valid, probabilities must be non-negative and sum to 1.
02

Evaluate Distribution (a)

For distribution (a), check if the probabilities sum to 1. Calculate: \[ P(0) + P(1) + P(2) = 0.25 + 0.60 + 0.15 = 1.00 \]Since the sum is 1, the distribution is valid.
03

Evaluate Distribution (b)

For distribution (b), check if the probabilities sum to 1.Calculate:\[ P(0) + P(1) + P(2) = 0.25 + 0.60 + 0.20 = 1.05 \]Since the sum is greater than 1, the distribution is not valid.

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Key Concepts

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

Random Variables
Random variables are a foundational concept in probability theory. Imagine a random variable as a function that assigns a numerical value to each outcome in a sample space. For instance, if you roll a six-sided die, a random variable could be the number that appears on the top face. Here's how it works:
  • A random variable can be discrete or continuous.
  • Discrete: Takes on a countable number of distinct values, like the possible outcomes of a die roll.
  • Continuous: Can take any value in a certain interval or range, such as measuring the time it takes for a bus to arrive.
Understanding how a random variable links possible outcomes to probabilities is key to comprehending probability distributions.
Probability Sum
The probability sum is a crucial condition in determining the validity of a probability distribution. It simply means that the probabilities of all possible outcomes of a random variable must add up to 1. This ensures that one of these outcomes is certain to occur. Consider this:
  • If you have a set of probabilities \{P(x_1), P(x_2), P(x_3), ..., P(x_n)\}, the mathematical expression requires: \[ \sum_{i=1}^{n} P(x_i) = 1 \]
  • This is the essence of a probability sum, ensuring the completeness of the entire sample space.
When evaluating if a distribution is valid, first check if the sum equals 1. This step is non-negotiable in probability theory.
Valid Distribution
A valid probability distribution fulfills specific conditions, ensuring that it accurately represents a statistical scenario. Two essential criteria must be met:
  • Sum of probabilities: Must equal 1, confirming all possible outcomes are accounted for.
  • Non-negative probabilities: Each probability must be zero or positive.
To understand it practically, evaluate distribution examples:
  • For distribution (a), where probabilities of outcomes 0, 1, and 2 are 0.25, 0.60, and 0.15 respectively, adding these gives 1, indicating it's valid.
  • For distribution (b), with probabilities 0.25, 0.60, and 0.20, the sum reaches 1.05, meaning it cannot represent a valid probability distribution since it exceeds 1.
Correctly evaluating the validity of a distribution helps ensure accurate probability modeling.
Non-negative Probabilities
Non-negative probabilities refer to the mathematical rule that each probability value in a distribution must be zero or greater. This is because negative probabilities are not logically possible in the real world. Think about the following:
  • Probabilities are a measure of likelihood, ranging from impossible (0) to certain (1).
  • Negative values would imply a less-than-impossible event, which doesn't make sense.
When forming a probability distribution, each assigned probability must be within the interval [0, 1]. Double-checking that all probabilities meet this condition needs to be part of the verification process for assessing distribution validity. Following this essential guideline helps maintain the integrity of any probabilistic model.

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Most popular questions from this chapter

Blood type A occurs in about \(41 \%\) of the population (Reference: Laboratory and Diagnostic Tests by F. Fischbach). A clinic needs 3 pints of type A blood. A donor usually gives a pint of blood. Let \(n\) be a random variable representing the number of donors needed to provide 3 pints of type A blood. (a) Explain why a negative binomial distribution is appropriate for the random variable \(n .\) Write out the formula for \(P(n)\) in the context of this application. Hint: See Problem 30. (b) Compute \(P(n=3), P(n=4), P(n=5),\) and \(P(n=6)\) (c) What is the probability that the clinic will need from three to six donors to obtain the needed 3 pints of type A blood? (d) What is the probability that the clinic will need more than six donors to obtain 3 pints of type A blood? (e) What are the expected value \(\mu\) and standard deviation \(\sigma\) of the random variable \(n ?\) Interpret these values in the context of this application.

A large bank vault has several automatic burglar alarms. The probability is 0.55 that a single alarm will detect a burglar. (a) How many such alarms should be used for \(99 \%\) certainty that a burglar trying to enter will be detected by at least one alarm? (b) Suppose the bank installs nine alarms. What is the expected number of alarms that will detect a burglar?

The following is based on information taken from The Wolf in the Southwest: The Making of an Endangered Species, edited by David Brown (University of Arizona Press). Before \(1918,\) approximately \(55 \%\) of the wolves in the New Mexico and Arizona region were male, and \(45 \%\) were female. However, cattle ranchers in this area have made a determined effort to exterminate wolves. From 1918 to the present, approximately \(70 \%\) of wolves in the region are male, and \(30 \%\) are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (a) Before \(1918,\) in a random sample of 12 wolves spotted in the region, what was the probability that 6 or more were male? What was the probability that 6 or more were female? What was the probability that fewer than 4 were female? (b) Answer part (a) for the period from 1918 to the present.

Suppose you are a hospital manager and have been told that there is no need to worry that respirator monitoring equipment might fail because the probability any one monitor will fail is only \(0.01 .\) The hospital has 20 such monitors and they work independently. Should you be more concerned about the probability that exactly one of the 20 monitors fails, or that at least one fails? Explain.

Consider a binomial experiment with \(n=7\) trials where the probability of success on a single trial is \(p=0.30\) (a) Find \(P(r=0)\) (b) Find \(P(r \geq 1)\) by using the complement rule.
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