Problem 219 - Applied Statistics and Probability for Engineers

Question

Suppose that the construction of a solar power station is initiated. The project's completion time has not been set due to uncertainties in financial resources. The proportion of completion within one year has a beta distribution with parameters \(\alpha=1\) and \(\beta=5 .\) Determine the following: (a) Mean and variance of the proportion completed within one year (b) Probability that more than half of the project is completed within one year (c) Proportion of the project that is completed within one year with probability 0.9

Step-by-Step Solution

Answer

(a) Mean is \( \frac{1}{6} \); variance is 0.0198. (b) Probability of more than half completion: 0.0313. (c) Proportion completed with 90% probability: 0.44.

Step 1: Understanding the Beta Distribution

The beta distribution is defined for a random variable \(X\) on the interval [0, 1] with probability density function given by \( f(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)} \), where \(\alpha > 0\) and \(\beta > 0\) are shape parameters and \(B(\alpha, \beta)\) is the beta function.

Step 2: Find Mean and Variance

The mean of a beta distribution is given by \( E(X) = \frac{\alpha}{\alpha + \beta} \) and the variance is given by \( Var(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} \). For \( \alpha = 1 \) and \( \beta = 5 \), we calculate \( E(X) = \frac{1}{6} \) and \( Var(X) = \frac{5}{252} \approx 0.0198 \).

Step 3: Calculate Probability that More Than Half is Completed

To find the probability \( P(X > 0.5) \), use the cumulative distribution function (CDF) for a beta distribution. The probability is given by \( 1 - F(0.5; \alpha, \beta) \). Using \( \alpha = 1 \) and \( \beta = 5 \), the CDF value can be found using statistical tables or software. The result is approximately 0.0313.

Step 4: Determine the Proportion Completed with 0.9 Probability

To find the 0.9 quantile of the beta distribution, where \( P(X \leq x) = 0.9 \), we can use statistical software or beta distribution tables. Solve for \( x \) where the CDF value equals 0.9. The quantile for \( \alpha = 1 \) and \( \beta = 5 \) is approximately 0.44.

Key Concepts

Mean of Beta Distribution

The mean of a beta distribution provides an average value of the distribution which helps us to understand how a beta-distributed project might behave. This is particularly useful when dealing with situations involving proportions, such as tracking the completion of a project over a given timeframe.

For a beta distribution with parameters \( \alpha \) and \( \beta \), the mean (or expected value) can be calculated using the formula: \[ E(X) = \frac{\alpha}{\alpha + \beta} \] This formula takes into account both shape parameters, which define the distribution. It's the ratio of \( \alpha \) (indicating the number of successes) over the total number of trials \( \alpha + \beta \).

When dealing with specific values, such as \( \alpha = 1 \) and \( \beta = 5 \), the mean becomes: \[ E(X) = \frac{1}{1 + 5} = \frac{1}{6} \] This indicates that, on average, the proportion of completion of the project within one year is expected to be about \(16.67\%\).

Understanding the mean gives us a benchmark against which to measure other quantities in the beta distribution.

Variance of Beta Distribution

The variance of a beta distribution tells us how much variability there is in the data around the mean. For project completion rates, this indicates how predictable the completion proportion is relative to the expected value.

The variance of a beta distribution with parameters \( \alpha \) and \( \beta \) is given by the formula: \[ Var(X) = \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} \] This formula measures the spread of the data in the distribution, with higher variance indicating more variability.

Plugging in \( \alpha = 1 \) and \( \beta = 5 \), the variance is calculated as: \[ Var(X) = \frac{1 \times 5}{(1 + 5)^2(1 + 5 + 1)} = \frac{5}{252} \approx 0.0198 \] This result suggests a low level of variability in the proportion of completion rate within the first year, indicating that the actual completion rates are likely to be close to the mean of \( \frac{1}{6} \).

Understanding variance in the context of beta distribution is crucial for assessing the risk and consistency of the outcomes expected.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a beta distribution provides valuable information about the probability that a random variable will take on a value less than or equal to a certain level. This function helps us quantify uncertainty by calculating probabilities for specific outcomes.

For the beta distribution, the CDF, \( F(x; \alpha, \beta) \), accumulates the probability up to a given value \( x \). It illustrates how likely it is for a project to be completed to certain proportions within a given timeframe.

To determine the probability that more than half of a project is completed within a year, we use the CDF by calculating \( P(X > 0.5) \), which is \( 1 - F(0.5; \alpha, \beta) \). With parameters \( \alpha = 1 \) and \( \beta = 5 \), this calculation uses software or statistical tables to yield approximately: \[ P(X > 0.5) = 1 - F(0.5; 1, 5) \approx 0.0313 \] This shows there's only a 3.13% chance that more than half of the project is completed within the year.

Utilizing the CDF allows you to understand where most of the distribution lies, which is very beneficial in project management for planning and risk assessment.

Quantile of Beta Distribution

Quantiles in a beta distribution help identify certain cut-points below which lies a specific proportion of the distribution. They are particularly useful for establishing thresholds or benchmarks, such as defining project milestones.

To find the quantile at a particular probability level, say 0.9, you want to determine the value of \( x \) such that \( P(X \leq x) = 0.9 \). This is effectively the value below which 90% of the distribution falls.

For a beta distribution with parameters \( \alpha = 1 \) and \( \beta = 5 \), you would employ numerical methods or statistical software to solve this for \( x \). The estimated 0.9 quantile for this beta distribution configuration is approximately 0.44.

In the context of project completion, this means there is a 90% chance that up to 44% of the project will be completed within one year. Understanding quantiles helps in setting realistic goals and expectations.