Question

Suppose the lifetime in months of an engine, working under hazardous conditions, has a \(\Gamma\) distribution with a mean of 10 months and a variance of 20 months squared. (a) Determine the median lifetime of an engine. (b) Suppose such an engine is termed successful if its lifetime exceeds 15 months. In a sample of 10 engines, determine the probability of at least 3 successful engines.

Short answer

The median lifetime of an engine is approximately 9.7 months. The probability of having at least 3 successful engines (assuming the afore-calculated \(p\) value) can be obtained using the binomial distribution theorem given \(n=10\), and \(k≥3\).

Step-by-step solution

Calculate Gamma Function Parameters

Since we know that the mean of a Gamma distribution is \(k \times θ\) and the variance is \(k \times θ^2\), we can use the provided mean and variance to solve this system of equations. Given: Mean=10 months and Variance=20 month squared. Solving these equations gives: \(k = θ\) and \(k = 10\). Thus, both \(k\) and \(θ\) equal to 10.

Determine the Median Lifetime

Gamma distribution does not have a simple closed-form solution to calculate median directly. So, we're going to approximate the median by assuming that our distribution is close to a normal distribution because the shape parameter \(k\) is large (10 in this case). For large \(k\), the gamma distribution is approximately normally distributed. Hence, the median is approximately equal to \((k-1/3)θ = (10 - 1/3) \times 10 = 9.7\) months.

Calculate Probability of Success

A engine is termed successful if its lifetime exceeds 15 months. First, we need to find probability \(p\) of an engine being successful. This equals one minus the cumulative distribution function value for 15, i.e., \(p = 1 - \ΓCDF(15; k=10, θ=10)\). Now, we want at least 3 successful engines out of 10, we apply the binomial theorem with \(p\), \(n=10\), and \(k≥3\) to obtain the probability.

Key concepts

Gamma Distribution Parameters

Understanding the parameters of the Gamma distribution is crucial for interpreting its characteristics and applications in statistics. In essence, the Gamma distribution is a two-parameter family of continuous probability distributions. The two key parameters are called the shape parameter, denoted by \(k\) (sometimes referred to as \(\alpha\)), and the scale parameter, denoted by \(\theta\) (sometimes referred to as \(\beta\)).

Let's break down these parameters:

• The shape parameter \(k\) determines the form of the distribution. When \(k\) is an integer, the distribution represents the sum of \(k\) exponentially distributed random variables, each with mean \(\theta\).
• The scale parameter \(\theta\) stretches or compresses the distribution horizontally along the x-axis and influences the spread of the data.

In the context of an engine's lifetime under hazardous conditions, the mean lifetime, which equals \(k \times \theta\), is 10 months, and the variance, \(k \times \theta^2\), is 20 months squared. Solving these equations indicates that both \(k\) and \(\theta\) equal 10 in this particular scenario, providing us details on how the lifetime of these engines is distributed.

Median of Gamma Distribution

The median of a Gamma distribution is the value that divides the probability density function into two equal areas, with half the values falling below the median and the other half above. Determining the exact median for a Gamma distribution can be complex because there is no simple closed-form solution. However, for large shape parameters, the Gamma distribution approaches a normal distribution due to the Central Limit Theorem.

In practical terms, when dealing with a large shape parameter like in our example where \(k=10\), we can approximate the median. The calculation often involves subtracting \(1/3\) from the shape parameter and then multiplying by the scale parameter, yielding an estimate for the median lifetime that is \((k-1/3) \theta = (10 - 1/3) \times 10 = 9.7\) months. Although this is an approximation, it is a helpful method to estimate the median when direct calculation is not feasible.

Probability with Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. This distribution is characterized by two parameters:

• The number of trials, \(n\), and
• The probability of success in a single trial, \(p\).

When we apply the concept of the binomial distribution to our engine lifetime problem, we are interested in calculating the probability of having a certain number of successful engines (lifetimes exceeding 15 months) out of a sample of 10.

First, we compute the probability \(p\) of a single engine being deemed successful, and then we use this probability in the binomial formula to determine the probability of at least 3 out of 10 engines exceeding the 15-month lifetime threshold. The binomial formula sums the probabilities of having exactly 3, 4, ..., 10 successful engines. This computational process enables us to understand how likely it is to observe a particular number of successes in a sample, which is a fundamental aspect of probability and statistics.