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Chapter 4: Problem 117

Use the applet Comparison of Beta Density Functions to compare beta density functions with \((\alpha=9, \beta=7),(\alpha=10, \beta=7),\) and \((\alpha=12, \beta=7)\). a. Are these densities symmetric? Skewed left? Skewed right? b. What do you observe as the value of \(\alpha\) gets closer to \(12 ?\) c. Graph some more beta densities with \(\alpha>1, \beta>1,\) and \(\alpha>\beta .\) What do you conjecture about the shape of beta densities with \(\alpha>\beta\) and both \(\alpha>1\) and \(\beta>1 ?\)

Short Answer

Expert verified
These densities are skewed left. As \(\alpha\) increases, left skew decreases. Beta densities with \(\alpha > \beta > 1\) are skewed left.

Step by step solution

01

Understanding Beta Distribution

The Beta distribution is defined by two shape parameters \(\alpha\) and \(\beta\). When \(\alpha > \beta\), the distribution is skewed to the left. Conversely, when \(\alpha < \beta\), it is skewed to the right. If \(\alpha = \beta\), it is symmetric.
02

Analysis of Given Densities

We have three beta densities: \((\alpha=9, \beta=7), (\alpha=10, \beta=7),\) and \((\alpha=12, \beta=7)\). Since \(\alpha > \beta\) for all, these are all skewed to the left.
03

Observing the Effect of Increasing \(\alpha\)

As \(\alpha\) increases from 9 to 12 while \(\beta\) remains constant, the left skew decreases, moving closer to a uniform distribution. The concentration of probability mass shifts to the right.
04

Conjecture from Additional Graphs

When plotting more beta densities with both \(\alpha > 1\) and \(\beta > 1\), and particularly where \(\alpha > \beta\), the distributions are generally skewed left. As \(\alpha\) increases significantly compared to \(\beta\), the distribution becomes more concentrated on the right.

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Key Concepts

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

Skewness
Skewness refers to the asymmetry observed in the probability distribution of a random variable. It essentially tells us whether all the probability mass leans towards one side of the distribution's central value or the other. The shape of a distribution can be classified by its skewness:
\(\alpha > \beta\) indicates a left skew (negative skewness) because the tail on the left side of the distribution is longer or fatter than the right side.
\(\alpha < \beta\) indicates a right skew (positive skewness) due to the tail on the right being longer than the left.
When \(\alpha = \beta\), the distribution is symmetric, meaning it has no skew.Skewness is an important factor in analyzing the characteristics of a data set because it affects interpretation. In the context of the Beta distribution, adjusting the values of \(\alpha\) and \(\beta\) changes the skewness and hence the portrayal of the dataset's tendencies.
Probability Density Function
The Probability Density Function (PDF) is a vital component of continuous probability distributions. It describes the likelihood of a random variable falling within a particular range of values. The area under the PDF between two values indicates the probability that the random variable falls within that range. For Beta distributions, the PDF is defined as:\[ f(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)} \]where \(B(\alpha, \beta)\) is the Beta function, ensuring the area under the curve integrates to 1. This formula shows how values close to 0 or 1 can have higher likelihoods depending on \(\alpha\) and \(\beta\).
The PDF aids in understanding the distribution's characteristics, enabling clear predictions about outcomes and tendencies within a given dataset.
Shape Parameters
The shape parameters \(\alpha\) and \(\beta\) in the Beta distribution are crucial because they determine the distribution's form and behavior.- **\(\alpha\)**: This parameter affects the concentration of probability mass near 1. Larger \(\alpha\) values pull the mass towards the end of the interval, decreasing skewness towards the left when \(\alpha > \beta\).- **\(\beta\)**: Influences the concentration near 0. As \(\beta\) increases, it pulls probability mass towards the start of the interval.Both parameters should be greater than 0. When both \(\alpha\) and \(\beta\) are greater than 1, and especially when \(\alpha > \beta\), they create distributions that are skewed to the left, often with a single peak. This configuration reflects scenarios where outcomes are concentrated towards higher values with a gradually tapering left tail. Understanding these parameters helps in modeling processes where occurrence probabilities need to reflect observed or expected real-world behaviors.

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Most popular questions from this chapter

If \(Y\) is a continuous random variable with density function \(f(y)\) that is symmetric about 0 (that is, \(f(y)=f(-y)\) for all \(y\) ) and \(E(Y)\) exists, show that \(E(Y)=0 .\) [Hint: \(E(Y)=\int_{-\infty}^{0} y f(y) d y+\int_{0}^{\infty} y f(y) d y .\) Make the change of variable \(w=-y\) in the first integral.]

Suppose that \(Y_{1}\) and \(Y_{2}\) are binomial random variables with parameters \(\left(n, p_{1}\right)\) and \(\left(n, p_{2}\right)\) respectively, where \(p_{1}\left(Y_{2}=0\right)\) b. Use the relationship between the beta distribution function and sums of binomial probabilities given in Exercise 4.134 to deduce that, if \(k\) is an integer between 1 and \(n-1\). $$P\left(Y_{1} \leq k\right)=\sum_{i=0}^{n}\left(\begin{array}{l}n \\\i\end{array}\right)\left(p_{1}\right)^{i}\left(1-p_{1}\right)^{n-i}=\int_{p_{1}}^{1} \frac{t^{k}(1-t)^{n-k-1}}{B(k+1, n-k)} d t$$. c. If \(k\) is an integer between 1 and \(n-1\), the same argument used in part (b). yields that $$P\left(Y_{2} \leq k\right)=\sum_{i=0}^{n}\left(\begin{array}{l} n \\\i\end{array}\right)\left(p_{2}\right)^{i}\left(1-p_{2}\right)^{n-i}=\int_{p_{2}}^{1} \frac{t^{k}(1t)^{n-k-1}}{B(k+1, n-k)} d t$$,Show that, if \(k\) is any integer between 1 and \(n-1, P\left(Y_{1} \leq k\right)>P\left(Y_{2} \leq k\right)\). Interpret this result.

A circle of radius \(r\) has area \(A=\pi r^{2} .\) If a random circle has a radius that is uniformly distributed on the interval \((0,1),\) what are the mean and variance of the area of the circle?

The change in depth of a river from one day to the next, measured (in feet) at a specific location, is a random variable \(Y\) with the following density function: $$ f(y)=\left\\{\begin{array}{ll} k, & -2 \leq y \leq 2 \\ 0, & \text { elsewhere } \end{array}\right. $$ a. Determine the value of \(k\). b. Obtain the distribution function for \(Y\).

Prove that the variance of a beta-distributed random variable with parameters \(\alpha\) and \(\beta\) is $$\sigma^{2}=\frac{\alpha \beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$$
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