Problem 185 - Applied Statistics and Probability for Engineers

Question

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

Step-by-Step Solution

Answer

(a) \(P(X<0.25) \approx 0.1422\); (b) \(P(0.25

Step 1: Understand the Beta Distribution

The Beta distribution is a continuous probability distribution defined over the interval \([0, 1]\), and it is parameterized by two positive shape parameters, \(\alpha\) and \(\beta\). Here, \(X\) follows a Beta distribution with \(\alpha = 2.5\) and \(\beta = 1\).

Step 2: Probability Density Function

The probability density function (PDF) of 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, \[ B(\alpha, \beta) = \int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \, dt \] Using these descriptions, we note this PDF formula is needed mainly for comprehension since integration will commonly use standard statistical tables or technology.

Step 3: Calculate P(X<0.25)

The cumulative distribution function (CDF) for the Beta distribution gives us \(P(X < a)\), and it is \[ F(a; \alpha, \beta) = \int_0^a f(x; \alpha, \beta) \, dx \]Using statistical tables or a calculator, find that:\[ P(X < 0.25) = I_{0.25}(2.5, 1) \approx 0.1422 \] where \(I_x(\alpha, \beta)\) is the regularized incomplete beta function.

Step 4: Calculate P(0.25 < X < 0.75)

Using the CDF to find the probability of an interval:\[ P(0.25 < X < 0.75) = P(X < 0.75) - P(X < 0.25) \]Use statistical tools to find:\[ P(X < 0.75) = I_{0.75}(2.5, 1) \approx 0.7981 \]Thus:\[ P(0.25 < X < 0.75) = 0.7981 - 0.1422 \approx 0.6559 \]

Step 5: Mean of the Beta Distribution

The mean of a Beta distribution is given by\[ \text{Mean} = \frac{\alpha}{\alpha + \beta} \]For \(\alpha = 2.5\) and \(\beta = 1\), the mean is:\[ \text{Mean} = \frac{2.5}{2.5 + 1} = \frac{2.5}{3.5} = \frac{5}{7} \approx 0.7143 \]

Step 6: Variance of the Beta Distribution

The variance of a Beta distribution is given by\[ \text{Variance} = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]Here, it becomes:\[ \text{Variance} = \frac{2.5 \cdot 1}{(2.5 + 1)^2 (2.5 + 1 + 1)} = \frac{2.5}{3.5^2 \cdot 4.5} \approx 0.0510 \]

Key Concepts

Probability Density Function

In a Beta Distribution, the Probability Density Function (PDF) plays a crucial role in understanding how probabilities are distributed within the range of possible outcomes. The PDF of a Beta distribution is expressed as \[ f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)} \]where \( B(\alpha, \beta) \)is the Beta function:\[ B(\alpha, \beta) = \int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \, dt \].
This PDF formula gives the likelihood of each outcome happening. Practically, for most calculations, knowing how to find or use pre-calculated values from tables or using software is more common, but understanding the formula aids deep comprehension.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) of a Beta distribution provides cumulative probability up to a certain point. For Beta Distribution, the CDF is defined as:\[F(a; \alpha, \beta) = \int_0^a f(x; \alpha, \beta) \, dx \,.\]
This means the CDF will give you the probability of a variable being less than or equal to that value \(a\). In practice, the actual integration is often performed using statistical tables or computational tools. For instance, given \(\alpha = 2.5\) and \(\beta = 1\), to find out the probability \(P(X < 0.25)\), you compute \(I_{0.25}(\alpha, \beta)\), which uses the regularized incomplete beta function.

Mean and Variance

The mean and variance are two important measures of a probability distribution's central tendency and dispersion, respectively. For a Beta Distribution, these values can be conveniently determined from its parameters \(\alpha\) and \(\beta\).

• The mean is calculated as:\[ \text{Mean} = \frac{\alpha}{\alpha + \beta} \]
• The variance is:\[ \text{Variance} = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]

These formulas help us understand where the data is likely to "cluster" and how spread out the values are. Using \(\alpha = 2.5\) and \(\beta = 1\), the mean and variance come out to approximately 0.7143 and 0.0510, respectively.

Regularized Incomplete Beta Function

The Regularized Incomplete Beta Function is a special mathematical function used in calculating probabilities for the Beta distribution. Denoted by\(I_x(\alpha, \beta)\), it represents the CDF of the Beta distribution, providing the probability that a Beta-distributed random variable \(X\) is less than or equal to \(x\).
This function simplifies the otherwise complex integration involved in the CDF calculation, making it more accessible for practical applications. It is widely available in statistical tables and in software packages that can handle advanced mathematical computations. Understanding \(I_x(\alpha, \beta)\) ensures one can efficiently handle Beta distribution problems in practical scenarios, particularly when working with non-integer parameters.

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Problem 186

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Key Concepts

Probability Density Function

In a Beta Distribution, the Probability Density Function (PDF) plays a crucial role in understanding how probabilities are distributed within the range of possible outcomes. The PDF of a Beta distribution is expressed as \[ f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)} \]where \( B(\alpha, \beta) \)is the Beta function:\[ B(\alpha, \beta) = \int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \, dt \].
This PDF formula gives the likelihood of each outcome happening. Practically, for most calculations, knowing how to find or use pre-calculated values from tables or using software is more common, but understanding the formula aids deep comprehension.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) of a Beta distribution provides cumulative probability up to a certain point. For Beta Distribution, the CDF is defined as:\[F(a; \alpha, \beta) = \int_0^a f(x; \alpha, \beta) \, dx \,.\]
This means the CDF will give you the probability of a variable being less than or equal to that value \(a\). In practice, the actual integration is often performed using statistical tables or computational tools. For instance, given \(\alpha = 2.5\) and \(\beta = 1\), to find out the probability \(P(X < 0.25)\), you compute \(I_{0.25}(\alpha, \beta)\), which uses the regularized incomplete beta function.

Mean and Variance

The mean and variance are two important measures of a probability distribution's central tendency and dispersion, respectively. For a Beta Distribution, these values can be conveniently determined from its parameters \(\alpha\) and \(\beta\).

• The mean is calculated as:\[ \text{Mean} = \frac{\alpha}{\alpha + \beta} \]
• The variance is:\[ \text{Variance} = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]

These formulas help us understand where the data is likely to "cluster" and how spread out the values are. Using \(\alpha = 2.5\) and \(\beta = 1\), the mean and variance come out to approximately 0.7143 and 0.0510, respectively.

Regularized Incomplete Beta Function

The Regularized Incomplete Beta Function is a special mathematical function used in calculating probabilities for the Beta distribution. Denoted by\(I_x(\alpha, \beta)\), it represents the CDF of the Beta distribution, providing the probability that a Beta-distributed random variable \(X\) is less than or equal to \(x\).
This function simplifies the otherwise complex integration involved in the CDF calculation, making it more accessible for practical applications. It is widely available in statistical tables and in software packages that can handle advanced mathematical computations. Understanding \(I_x(\alpha, \beta)\) ensures one can efficiently handle Beta distribution problems in practical scenarios, particularly when working with non-integer parameters.