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Magazine Chapter 4: Problem 1
Suppose that \(x\) has a beta distribution with parameters \(\alpha=2.5\) and
\(\beta=1 .\) Determine the following:
a. \(P(X<0.25)\)
b. \(P(0.25
Short Answer
Expert verified
a. 0.125
b. 0.71875
c. Mean: 0.7143, Variance: 0.051
Step by step solution
01
Understanding the Beta Distribution
A beta distribution with parameters \(\alpha\) and \(\beta\) is a continuous probability distribution over the interval \([0, 1]\). The probability density function (PDF) for a beta distribution is given by: \[ f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)} \] where \(B(\alpha, \beta)\) is the beta function. In this problem, \(\alpha = 2.5\) and \(\beta = 1\).
02
Calculating P(X
The cumulative distribution function (CDF) for a beta distribution is defined using the incomplete beta function. For \(P(X<0.25)\), this is written as \(F(0.25; 2.5, 1) = I_{0.25}(2.5, 1)\), where \(I_x(\alpha, \beta)\) is the regularized incomplete beta function. Using computational tools or tables, we find: \(P(X<0.25) \approx 0.125\).
03
Calculating P(0.25
To find \(P(0.25<X<0.75)\), we calculate the difference \(F(0.75; 2.5, 1) - F(0.25; 2.5, 1)\). Using the regularized incomplete beta function again, we get \(F(0.75; 2.5, 1) \approx 0.84375\). Thus, \(P(0.25<X<0.75) = 0.84375 - 0.125 = 0.71875\).
04
Calculating the Mean
For a beta distribution, the mean \(\mu\) is calculated using the formula: \(\mu = \frac{\alpha}{\alpha + \beta}\). Substituting the values \(\alpha = 2.5\) and \(\beta = 1\), we get \(\mu = \frac{2.5}{3.5} \approx 0.7143\).
05
Calculating the Variance
For a beta distribution, the variance \(\sigma^2\) is given by: \(\sigma^2 = \frac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\). Substituting the values \(\alpha = 2.5\) and \(\beta = 1\), we find \(\sigma^2 = \frac{2.5 \times 1}{3.5^2 \times 4.5} \approx 0.051\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Probability Density Function (PDF)
In a beta distribution, the probability density function (PDF) helps determine the likelihood of different outcomes within the interval [0, 1]. The PDF is expressed as follows: \( f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)} \), where \(B(\alpha, \beta)\) is the beta function.
- \(\alpha\) and \(\beta\) are shape parameters that influence the skewness of the distribution.
- The function produces a curve that illustrates the probabilities of all potential outcomes in the given interval.
- In this exercise, \(\alpha = 2.5\) and \(\beta = 1\), creating a distribution curve that leans towards one side.
Exploring the Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) represents the probability that a beta-distributed random variable \(X\) is less than a certain value \(x\). It is expressed as \( F(x; \alpha, \beta) = I_x(\alpha, \beta) \), where \( I_x(\alpha, \beta) \) is the regularized incomplete beta function.
- The CDF accumulates the probabilities from 0 up to a value \(x\), providing a complete picture of the distribution up to that point.
- For example, \(P(X < 0.25)\) in the solution is assessed as 0.125 from the CDF.
- To find probabilities over an interval, like \(P(0.25 < X < 0.75)\), you subtract one CDF value from another, using the endpoints of the interval.
Mean and Variance of the Beta Distribution
In the beta distribution, the mean \(\mu\) and variance \(\sigma^2\) provide insights into the distribution's central tendency and how spread out the values are.
- The mean \(\mu\) is determined by \(\mu = \frac{\alpha}{\alpha + \beta}\). Here, \(\alpha = 2.5\) and \(\beta = 1\) give a mean of approximately 0.7143.
- Variance \(\sigma^2\) measures the distribution's spread. It's calculated as \(\sigma^2 = \frac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\), which equals about 0.051 in this case.
- The mean indicates the "average" point of the distribution, while the variance gives insights into its consistency - smaller variance suggests tighter clustering of data.
Incomplete Beta Function and Its Role
The incomplete beta function is fundamental in calculating probabilities within the beta distribution, specifically in CDF calculations. It is denoted as \( I_x(\alpha, \beta) \) and is a key component for solving problems related to these distributions.
- This function takes three parameters, \(x\), \(\alpha\), and \(\beta\), to provide cumulative probabilities up to a specified point \(x\).
- It is termed "incomplete" because it computes the integral of the beta density function from 0 to \(x\) rather than over the entire domain.
- For example, for \(P(X<0.25)\), we use \( I_{0.25}(2.5, 1) \), translating into a probability of about 0.125.
