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Chapter 4: Problem 121

Use the applet Comparison of Beta Density Functions to compare beta -density functions with \((\alpha=0.5, \beta=0.7),(\alpha=0.7, \beta=0.7),\) and \((\alpha=0.9, \beta=0.7)\). a. What is the general shape of these densities? b. What do you observe as the value of \(\alpha\) gets larger?

Short Answer

Expert verified
The densities are generally 'U' shaped. As \(\alpha\) increases, the distribution flattens.

Step by step solution

01

Understanding Beta Density Functions

The beta distribution is a continuous probability distribution with two positive shape parameters, \(\alpha\) and \(\beta\). The shape of the beta distribution depends on the values of these parameters. When \(\alpha\) and \(\beta\) are both less than 1, the distribution typically has a 'U' shape or is bimodal.
02

Analyze Beta Density with \(\alpha=0.5, \beta=0.7\)

For \(\alpha = 0.5\) and \(\beta = 0.7\), the beta density function generally has a more pronounced 'U' shape or bimodal distribution. This is because both parameters are less than 1, emphasizing the peaks near 0 and 1.
03

Analyze Beta Density with \(\alpha=0.7, \beta=0.7\)

For \(\alpha = 0.7\) and \(\beta = 0.7\), the density becomes less bimodal compared to \(\alpha = 0.5\). With both parameters closer to 1, the peaks near 0 and 1 are less pronounced, producing a distribution that flattens slightly towards a uniform distribution.
04

Analyze Beta Density with \(\alpha=0.9, \beta=0.7\)

As \(\alpha\) increases to 0.9 while maintaining \(\beta = 0.7\), the distribution continues to flatten. The peak near 0 becomes less steep, indicating a tendency towards rising more uniformly across the distribution, although this effect is moderate.
05

Observing the Effect of Increasing \(\alpha\)

With an increase in \(\alpha\), while \(\beta\) is held constant at 0.7, the distribution transitions from a more pronounced 'U' shape to a flatter, more uniform-like shape. This observation shows that as \(\alpha\) approaches \(\beta\), the extremes of the distribution become less pronounced and the distribution gradually flattens.

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Key Concepts

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

Probability Distribution
A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It tells us how the probabilities are distributed over the values of the random variable. In the context of the beta distribution, this distribution is particularly useful for representing random variables that are constrained within an interval, typically between 0 and 1. This makes the beta distribution a popular choice for modeling random variables like proportions or percentages.
  • Typically, a probability distribution can be discrete, dealing with countable outcomes, or continuous, dealing with a range of outcomes.
  • The beta distribution is a continuous type.
  • Understanding the distribution helps in determining how likely particular outcomes are.
Probability distributions allow us to visualize and analyze data, making them a fundamental concept in statistics and probability.
Shape Parameters
In the beta distribution, the shape of the distribution is determined by two parameters:
  • \(\alpha\) (Alpha)
  • \(\beta\) (Beta)
Both \(\alpha\) and \(\beta\) are known as shape parameters. They play a significant role in defining the form of the beta distribution curve.
  • If both parameters are less than 1, the distribution tends to be bimodal, showcasing peaks near 0 and 1.
  • As these parameters increase, particularly getting closer to each other, the distribution's shape becomes less extreme and can even resemble a uniform distribution.
  • When \(\alpha\) equals \(\beta\), the distribution is symmetric around 0.5.
These parameters enable flexibility in the beta distribution, allowing it to model a wide variety of shapes depending on their values.
Continuous Distribution
A continuous distribution, unlike a discrete distribution, can take an infinite number of possible outcomes within a given range. The beta distribution exemplifies a continuous distribution because it deals with all possible values between 0 and 1 over which the random variable may take any value.
  • It is effective in scenarios where outcomes are measured in intervals, like percentages or probabilities.
  • The continuous nature allows for a smooth curve which can model real-world phenomena.
  • Key characteristics such as mean, variance, and shape of the curve are dependent on the values of \(\alpha\) and \(\beta\).
The concept of continuity is crucial in modeling and simulating processes that have a continuum of states.
Bimodal Distribution
A bimodal distribution is characterized by having two distinct peaks. In the context of beta distribution, these peaks often appear when the shape parameters \(\alpha\) and \(\beta\) are both less than 1. This 'U' shape is created as the probability is concentrated towards the edges, near 0 and 1.
  • This property allows the beta distribution to be useful in cases where outcomes might gravitate towards extremes rather than a mean.
  • It's important to understand that a distribution can become bimodal based on the choice of its parameters.
  • Bimodal distributions are valuable for reflecting scenarios in which two dominant frequencies or modes appear in the data.
Understanding when a distribution is bimodal can reveal insight into the underlying process being studied, as it points towards a natural separation within the dataset.

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Most popular questions from this chapter

The cycle time for trucks hauling concrete to a highway construction site is uniformly distributed over the interval 50 to 70 minutes. What is the probability that the cycle time exceeds 65 minutes if it is known that the cycle time exceeds 55 minutes?

Suppose that \(Y\) has a beta distribution with parameters \(\alpha\) and \(\beta\). a. If \(a\) is any positive or negative value such that \(\alpha+a>0\), show that $$ E\left(Y^{a}\right)=\frac{\Gamma(\alpha+a) \Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\alpha+\beta+a)} $$ b. Why did your answer in part (a) require that \(\alpha+a>0\) ? c. Show that, with \(a=1,\) the result in part (a) gives \(E(Y)=\alpha /(\alpha+\beta)\) d. Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about \(\alpha ?\) e. Use the result in part (a) to give an expression for \(E(1 / Y), E(1 / \sqrt{Y})\), and \(E\left(1 / Y^{2}\right)\). What do you need to assume about \(\alpha\) in each case?

If \(\theta_{1}<\theta_{2}\), derive the moment-generating function of a random variable that has a uniform distribution on the interval \(\left(\theta_{1}, \theta_{2}\right)\)

A supplier of kerosene has a 150 -gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If \(Y\) denotes weekly demand in hundreds of gallons, the relative frequency of demand can be modeled by $$f(y)=\left\\{\begin{array}{ll}y, & 0 \leq y \leq 1, \\\1, & 1

Let \(Y\) have density function $$ f(y)=\left\\{\begin{array}{ll} c y e^{-2 y}, & 0 \leq y \leq \infty \\ 0, & \text { elsewhere } \end{array}\right. $$ a. Find the value of \(c\) that makes \(f(y)\) a density function. b. Give the mean and variance for \(Y\). c. Give the moment-generating function for \(Y\).
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