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Chapter 5: Problem 100

Verify whether or not each of the following is a probability function. State your conclusion and explain. a. \(f(x)=\frac{3 x}{8 x !},\) for \(x=1,2,3,4\) b. \(f(x)=0.125,\) for \(x=0,1,2,3,\) and \(f(x)=0.25\) for \(x=4,5\) c. \(f(x)=(7-x) / 28,\) for \(x=0,1,2,3,4,5,6,7\) d. \(f(x)=\left(x^{2}+1\right) / 60,\) for \(x=0,1,2,3,4,5\)

Short Answer

Expert verified
The verification of these functions as probability functions will rely on calculating the functions at given points and ensuring that every result is greater than or equal to 0 and also that the sum of these results is 1. The final conclusion will depend on the calculations of these functions at each given point.

Step by step solution

01

Check condition for function a

For \(f(x)=\frac{3 x}{8 x !}, x=1,2,3,4\): Evaluate the function at each point and add all the results. If the sum is 1 and each value is greater than or equal 0, then it is a valid probability function.
02

Check condition for function b

For \(f(x)=0.125, x=0,1,2,3\) and \(f(x)=0.25, x=4,5\): Validate that the probabilities are greater than or equal to 0. Next, verify that the total sum of function values equals 1 to confirm the validity of the probability function.
03

Check condition for function c

For \(f(x)=(7-x) / 28, x=0,1,2,3,4,5,6,7\): Calculate each function value to make sure all values are greater than or equal to 0. Then, also evaluate their sum and compare with 1 to decide whether this is a valid probability function.
04

Check condition for function d

For \(f(x)=(x^{2}+1) / 60, x=0,1,2,3,4,5\): Perform the same checks on all x-values, ensuring that the function produces results greater than or equal to 0. Sum all results and check if it is equals to 1.

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Key Concepts

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

Probability Distribution
In probability theory, a probability distribution is a function that describes the likelihood of different outcomes in an experiment. It's fundamental for understanding uncertain events. A probability distribution can demonstrate which values a random variable can take and the probability associated with each of these values.

A probability distribution must satisfy specific conditions:
  • Each probability value must be between 0 and 1. This is because probability signifies a likelihood, and cannot exceed certainty, represented as 1.
  • The sum of all probability values must be equal to 1. This ensures that one of the possible outcomes is certain to occur.
Understanding these properties is crucial when working with probability distributions as it determines if a given function can indeed be a probability function. Verifying if these conditions hold is the first step in probability function verification, enabling you to gauge if the distribution is valid.
Probability Function Verification
Verifying a probability function involves checking if a given function can serve as a valid probability distribution function. To do this, follow these simple steps:

Firstly, ensure that each outcome of the function produces a probability greater than or equal to zero. This ensures there are no negative probabilities, which would not make sense in real-world scenarios.

Secondly, calculate the sum of all function values. If the sum equals 1, then the distribution is complete, indicating that the function indeed represents a valid set of probabilities. If either condition fails, the function is not a probability function. For instance, consider the function from the exercise for different values of \(x\) to determine validity; if the summation deviates from 1 or contains a negative probability, it needs reassessment.
Discrete Probability Distributions
A discrete probability distribution applies to scenarios where the set of possible outcomes is finite or countably infinite. In simple terms, it deals with discrete random variables, such as the outcome of rolling a die or the number of defective items in a batch.

The function associated with a discrete probability distribution is often referred to as a probability mass function (PMF). Like other distributions, it must satisfy the foundational probability rules: each probability is between 0 and 1, and the total probability across all outcomes equals 1.

When analyzing the given functions in the original exercise, each must be checked against these conditions. You essentially treat each random variable outcome as an individual point and seek to validate the entire structure to determine if it's a suitable discrete probability distribution. This approach helps in practical applications like quality control or decision-making based on discrete outcomes.

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Most popular questions from this chapter

According to United Mileage Plus Visa (November22,2004)\(, 41 \%\) of passengers say they "put on the earphones" to avoid being bothered by their seatmates during flights. To show how important, or not important, the earphones are to people, consider the variable \(x\) to be the number of people in a sample of 12 who say they "put on the earphones" to avoid their seatmates. Assume the\(41 \%\) is true for the whole population of airline travelers and that a random sample is selected. a. Is \(x\) a binomial random variable? Justify your answer. b. Find the probability that \(x=4\) or 5. c. Find the mean and standard deviation of \(x .\) d. Draw a histogram of the distribution of \(x:\) label it completely, highlight the area representing \(x=4\) and \(x=5,\) draw a vertical line at the value of the mean, and mark the location of \(x\) that is 1 standard deviation larger than the mean.

Let \(x\) be a random variable with the following probability distribution: $$\begin{array}{l|cccc}\hline x & 0 & 1 & 2 & 3 \\\P(x) & 0.4 & 0.3 & 0.2 & 0.1 \\\\\hline \end{array}$$ Does \(x\) have a binomial distribution? Justify your answer.

The town council has nine members. A proposal to establish a new industry in this town has been tabled, and all proposals must have at least two-thirds of the votes to be accepted. If we know that two members of the town council are opposed and that the others randomly vote "in favor" and "against," what is the probability that the proposal will be accepted?

Use a computer to find the probabilities for all possible \(x\) values for a binomial experiment where \(n=30\) and \(p=0.35.\)

A binomial random variable is based on \(n=20\) and \(p=0.4 .\) Find \(\Sigma x^{2} P(x).\)
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