Joint PDF Of Sufficient Statistic Y1 = X1 + X2 And Y2 = X2 For Exponential Distribution

In statistical inference, the concept of sufficient statistics plays a crucial role in simplifying data analysis and parameter estimation. Sufficient statistics are functions of the sample data that capture all the information relevant to the parameter of interest. In simpler terms, if we know the value of a sufficient statistic, we don't need the original data to make inferences about the parameter. This significantly reduces the complexity of the analysis without sacrificing any information. To truly grasp the concept of sufficient statistics, it's essential to delve into their mathematical definition and properties. A statistic Y = g(X) is said to be sufficient for a parameter θ if the conditional distribution of the sample X given Y does not depend on θ. This means that once we know the value of Y, the original data X provides no additional information about θ. The factorization theorem provides a powerful tool for identifying sufficient statistics. It states that a statistic Y = g(X) is sufficient for θ if and only if the joint probability density function (p.d.f.) of the sample can be factored into two functions: one that depends on the sample X only through the statistic Y and another that depends on θ but not on X. Mathematically, this can be expressed as: f(x; θ) = g(y; θ) h(x), where g(y; θ) depends on the data only through the statistic Y and h(x) does not depend on θ. This theorem allows us to systematically find sufficient statistics by examining the structure of the joint p.d.f. of the sample. For instance, in the context of the exponential distribution, the sum of the observations is a sufficient statistic for the rate parameter, as we will demonstrate later in this article. Understanding sufficient statistics is vital for efficient statistical inference. By focusing on sufficient statistics, we can reduce the dimensionality of the data while preserving all the relevant information for parameter estimation and hypothesis testing. This leads to simpler and more computationally efficient statistical procedures. In the realm of parameter estimation, sufficient statistics are often used to construct minimum variance unbiased estimators (MVUEs). An MVUE is an estimator that has the smallest variance among all unbiased estimators. The Rao-Blackwell theorem provides a method for improving an estimator by conditioning it on a sufficient statistic. This theorem states that if we have an unbiased estimator of a parameter, we can obtain another unbiased estimator with a smaller variance by taking the conditional expectation of the original estimator given a sufficient statistic. This technique is widely used in statistical practice to obtain efficient estimators. Furthermore, sufficient statistics play a crucial role in hypothesis testing. The Neyman-Pearson lemma, a fundamental result in hypothesis testing, states that the most powerful test for testing a simple null hypothesis against a simple alternative hypothesis is based on the likelihood ratio, which involves the ratio of the likelihood functions evaluated under the null and alternative hypotheses. When a sufficient statistic exists, the likelihood ratio depends only on the sufficient statistic, simplifying the test procedure. In summary, sufficient statistics are essential tools for simplifying statistical inference. They allow us to reduce the dimensionality of the data, construct efficient estimators, and develop powerful hypothesis tests. Understanding the concept of sufficient statistics is crucial for anyone working with statistical data. By identifying and utilizing sufficient statistics, we can extract the maximum amount of information from the data with minimal effort.

The exponential distribution is a fundamental probability distribution in statistics that describes the time until an event occurs in a Poisson process, where events occur continuously and independently at a constant average rate. It is widely used to model various phenomena, such as the lifetime of electronic components, the duration of phone calls, and the time between customer arrivals at a service facility. The exponential distribution is characterized by a single parameter, θ, which represents the mean time between events. The probability density function (p.d.f.) of the exponential distribution is given by: f(x; θ) = (1/θ) e^(-x/θ), 0 < x < ∞, 0 < θ < ∞, zero elsewhere, where x represents the time until the event occurs. This p.d.f. shows that the probability density decreases exponentially as x increases, indicating that shorter times are more likely than longer times. The cumulative distribution function (c.d.f.) of the exponential distribution is given by: F(x; θ) = 1 - e^(-x/θ), 0 < x < ∞. This function gives the probability that the event occurs before time x. The exponential distribution has several key properties that make it a valuable tool in statistical modeling. One of the most important properties is its memoryless property. This means that the probability of an event occurring in the future does not depend on how much time has already passed. Mathematically, this can be expressed as: P(X > s + t | X > s) = P(X > t), where X is a random variable following an exponential distribution, and s and t are positive time values. This property is particularly useful in modeling situations where the past history does not affect future events, such as the failure of electronic components due to random shocks. Another important property of the exponential distribution is its relationship to the Poisson distribution. The exponential distribution describes the time between events in a Poisson process, while the Poisson distribution describes the number of events occurring in a fixed interval of time. If events occur according to a Poisson process with rate λ, then the time between events follows an exponential distribution with mean 1/λ. This connection between the two distributions allows us to model both the number of events and the time between events in a unified framework. The exponential distribution is also closely related to the gamma distribution. The gamma distribution is a generalization of the exponential distribution that allows for non-integer shape parameters. The exponential distribution is a special case of the gamma distribution with a shape parameter of 1. The gamma distribution is useful for modeling waiting times for multiple events in a Poisson process, while the exponential distribution models the waiting time for the first event. In statistical inference, the exponential distribution is often used to model data that exhibits exponential decay, such as the lifetime of products or the time until a machine failure. Parameter estimation for the exponential distribution typically involves estimating the rate parameter θ. The maximum likelihood estimator (MLE) for θ is the sample mean, which is the sum of the observations divided by the sample size. This estimator is unbiased and efficient, meaning that it has the smallest variance among all unbiased estimators. Hypothesis testing for the exponential distribution often involves testing hypotheses about the rate parameter θ. Common tests include the likelihood ratio test and the Wald test. These tests allow us to determine whether there is evidence to support a claim about the rate parameter based on the observed data. In summary, the exponential distribution is a versatile probability distribution with a wide range of applications in statistics and probability. Its memoryless property, relationship to the Poisson and gamma distributions, and well-established statistical inference procedures make it a valuable tool for modeling and analyzing data in various fields.

Now, let's dive into the specific problem at hand. We are given a random sample X₁, X₂ of size 2 from an exponential distribution with the probability density function (p.d.f.) f(x; θ) = (1/θ) e^(-x/θ), where 0 < x < ∞ and 0 < θ < ∞. Our primary goal is to determine the joint p.d.f. of the sufficient statistic Y₁ = X₁ + X₂ for θ and Y₂ = X₂. This involves several key steps, including finding the joint p.d.f. of the sample, identifying a sufficient statistic for θ, and then transforming the random variables to obtain the joint p.d.f. of the sufficient statistic and another variable. The first step is to find the joint p.d.f. of the sample X₁, X₂. Since X₁ and X₂ are independent random variables from the same exponential distribution, their joint p.d.f. is the product of their individual p.d.f.s: f(x₁, x₂; θ) = f(x₁; θ) * f(x₂; θ) = (1/θ) e^(-x₁/θ) * (1/θ) e^(-x₂/θ) = (1/θ²) e^(-(x₁+x₂)/θ), where 0 < x₁ < ∞ and 0 < x₂ < ∞. This joint p.d.f. describes the probability density of observing the pair (x₁, x₂) for a given value of θ. Next, we need to identify a sufficient statistic for θ. A sufficient statistic is a function of the sample data that contains all the information relevant to the parameter of interest. In this case, we want to find a function of X₁ and X₂ that captures all the information about θ. The factorization theorem provides a powerful tool for identifying sufficient statistics. It states that a statistic T(X) is sufficient for θ if the joint p.d.f. of the sample can be factored into two functions: one that depends on the sample X only through the statistic T(X) and another that depends on θ but not on X. Applying the factorization theorem to the joint p.d.f. of X₁ and X₂, we can rewrite it as: f(x₁, x₂; θ) = (1/θ²) e^(-(x₁+x₂)/θ) = (1/θ²) e^(-y₁/θ), where y₁ = x₁ + x₂. Here, we can see that the joint p.d.f. can be factored into two functions: (1/θ²) e^(-y₁/θ), which depends on the data only through the statistic Y₁ = X₁ + X₂, and 1, which does not depend on X₁ or X₂. Therefore, by the factorization theorem, Y₁ = X₁ + X₂ is a sufficient statistic for θ. This means that if we know the value of Y₁, we don't need the original data X₁ and X₂ to make inferences about θ. Now, we need to find the joint p.d.f. of Y₁ = X₁ + X₂ and Y₂ = X₂. This involves a transformation of random variables. We have two random variables, X₁ and X₂, and we want to find the joint distribution of two new random variables, Y₁ and Y₂. The standard technique for this is to use the transformation formula, which involves the Jacobian determinant of the transformation. The transformation formula states that if we have a transformation from (X₁, X₂) to (Y₁, Y₂) given by Y₁ = g₁(X₁, X₂) and Y₂ = g₂(X₁, X₂), then the joint p.d.f. of (Y₁, Y₂) is given by: f(y₁, y₂; θ) = f(x₁( y₁, y₂), x₂( y₁, y₂); θ) |J|, where |J| is the absolute value of the Jacobian determinant of the transformation. The Jacobian determinant is given by: J = det [∂(x₁, x₂)/∂(y₁, y₂)] = (∂x₁/∂y₁) (∂x₂/∂y₂) - (∂x₁/∂y₂) (∂x₂/∂y₁). In our case, we have Y₁ = X₁ + X₂ and Y₂ = X₂. Solving for X₁ and X₂ in terms of Y₁ and Y₂, we get X₂ = Y₂ and X₁ = Y₁ - Y₂. Now we can compute the Jacobian determinant. ∂x₁/∂y₁ = 1, ∂x₁/∂y₂ = -1, ∂x₂/∂y₁ = 0, ∂x₂/∂y₂ = 1. J = (1)(1) - (-1)(0) = 1. Therefore, |J| = 1. Now we can substitute X₁ = Y₁ - Y₂ and X₂ = Y₂ into the joint p.d.f. of X₁ and X₂ to obtain the joint p.d.f. of Y₁ and Y₂. f(y₁, y₂; θ) = (1/θ²) e^(-(y₁-y₂+y₂)/θ) |1| = (1/θ²) e^(-y₁/θ), where 0 < y₂ < y₁ < ∞. This is the joint p.d.f. of the sufficient statistic Y₁ = X₁ + X₂ and Y₂ = X₂. The region of support for Y₁ and Y₂ is determined by the constraints 0 < x₁ < ∞ and 0 < x₂ < ∞, which translate to 0 < y₂ < ∞ and 0 < y₁ - y₂ < ∞, or equivalently, 0 < y₂ < y₁ < ∞. In summary, we have found the joint p.d.f. of the sufficient statistic Y₁ = X₁ + X₂ and Y₂ = X₂. This joint p.d.f. is essential for making inferences about the parameter θ based on the sufficient statistic Y₁.

In this section, we will provide a step-by-step derivation of the joint probability density function (PDF) of Y₁ and Y₂. This involves a transformation of random variables from (X₁, X₂) to (Y₁, Y₂), where Y₁ = X₁ + X₂ and Y₂ = X₂. The derivation relies on the transformation formula, which relates the joint PDFs of the original and transformed variables through the Jacobian determinant. Recall that the joint PDF of X₁ and X₂ is given by: f(x₁, x₂; θ) = (1/θ²) e^(-(x₁+x₂)/θ), where 0 < x₁ < ∞ and 0 < x₂ < ∞. Our goal is to find the joint PDF of Y₁ and Y₂, denoted by f(y₁, y₂; θ). To do this, we will use the transformation formula, which states that: f(y₁, y₂; θ) = f(x₁( y₁, y₂), x₂( y₁, y₂); θ) |J|, where |J| is the absolute value of the Jacobian determinant of the transformation. The first step is to express X₁ and X₂ in terms of Y₁ and Y₂. We have the following transformation: Y₁ = X₁ + X₂, Y₂ = X₂. Solving for X₁ and X₂ in terms of Y₁ and Y₂, we get: X₂ = Y₂, X₁ = Y₁ - Y₂. Next, we need to compute the Jacobian determinant of the transformation. The Jacobian determinant is given by: J = det [∂(x₁, x₂)/∂(y₁, y₂)] = (∂x₁/∂y₁) (∂x₂/∂y₂) - (∂x₁/∂y₂) (∂x₂/∂y₁). We need to compute the partial derivatives: ∂x₁/∂y₁ = ∂(y₁ - y₂)/∂y₁ = 1, ∂x₁/∂y₂ = ∂(y₁ - y₂)/∂y₂ = -1, ∂x₂/∂y₁ = ∂(y₂)/∂y₁ = 0, ∂x₂/∂y₂ = ∂(y₂)/∂y₂ = 1. Now we can plug these partial derivatives into the formula for the Jacobian determinant: J = (1)(1) - (-1)(0) = 1. The absolute value of the Jacobian determinant is |J| = |1| = 1. Now we can substitute X₁ = Y₁ - Y₂ and X₂ = Y₂ into the joint PDF of X₁ and X₂: f(x₁( y₁, y₂), x₂( y₁, y₂); θ) = f(y₁ - y₂, y₂; θ) = (1/θ²) e^(-((y₁-y₂)+y₂)/θ) = (1/θ²) e^(-y₁/θ). Finally, we can apply the transformation formula to obtain the joint PDF of Y₁ and Y₂: f(y₁, y₂; θ) = f(x₁( y₁, y₂), x₂( y₁, y₂); θ) |J| = (1/θ²) e^(-y₁/θ) * 1 = (1/θ²) e^(-y₁/θ). This is the joint PDF of Y₁ and Y₂. However, we also need to determine the region of support for Y₁ and Y₂. The region of support for X₁ and X₂ is 0 < x₁ < ∞ and 0 < x₂ < ∞. We need to translate these inequalities into inequalities for Y₁ and Y₂. From X₂ = Y₂, we have 0 < y₂ < ∞. From X₁ = Y₁ - Y₂, we have 0 < y₁ - y₂ < ∞, which implies y₂ < y₁. Combining these inequalities, we get the region of support for Y₁ and Y₂: 0 < y₂ < y₁ < ∞. Therefore, the joint PDF of Y₁ and Y₂ is given by: f(y₁, y₂; θ) = (1/θ²) e^(-y₁/θ), for 0 < y₂ < y₁ < ∞, zero elsewhere. This completes the derivation of the joint PDF of Y₁ and Y₂. The key steps involved expressing the original variables in terms of the transformed variables, computing the Jacobian determinant, substituting into the transformation formula, and determining the region of support for the transformed variables.

Having derived the joint probability density function (p.d.f.) of Y₁ and Y₂, we can now explore its implications and consider further analysis. The joint p.d.f. provides valuable information about the relationship between the sufficient statistic Y₁ and the auxiliary variable Y₂. It also allows us to derive the marginal distributions of Y₁ and Y₂ and investigate their statistical properties. One of the key observations from the joint p.d.f. f(y₁, y₂; θ) = (1/θ²) e^(-y₁/θ) is that it depends on the parameter θ only through Y₁. This is consistent with the fact that Y₁ is a sufficient statistic for θ. The variable Y₂ provides no additional information about θ beyond what is already contained in Y₁. To gain a deeper understanding of the behavior of Y₁ and Y₂, it is useful to derive their marginal distributions. The marginal p.d.f. of Y₁ can be obtained by integrating the joint p.d.f. over all possible values of Y₂: f(y₁; θ) = ∫ f(y₁, y₂; θ) dy₂. In our case, the limits of integration are 0 < y₂ < y₁, so we have: f(y₁; θ) = ∫₀^(y₁) (1/θ²) e^(-y₁/θ) dy₂ = (1/θ²) e^(-y₁/θ) ∫₀^(y₁) dy₂ = (1/θ²) e^(-y₁/θ) [y₂]₀^(y₁) = (1/θ²) e^(-y₁/θ) y₁, for 0 < y₁ < ∞. This shows that Y₁ follows a gamma distribution with shape parameter 2 and rate parameter 1/θ. The marginal p.d.f. of Y₂ can be obtained by integrating the joint p.d.f. over all possible values of Y₁: f(y₂; θ) = ∫ f(y₁, y₂; θ) dy₁. In our case, the limits of integration are y₂ < y₁ < ∞, so we have: f(y₂; θ) = ∫(y₂)^∞ (1/θ²) e^(-y₁/θ) dy₁ = (1/θ²) ∫(y₂)^∞ e^(-y₁/θ) dy₁ = (1/θ²) [-θ e^{(-y₁/θ)]_(y₂)}∞ = (1/θ²) [0 - (-θ e^(-y₂/θ))] = (1/θ) e^(-y₂/θ), for 0 < y₂ < ∞. This shows that Y₂ follows an exponential distribution with rate parameter 1/θ, which is the same distribution as the original sample X₁ and X₂. The marginal distributions of Y₁ and Y₂ provide insights into their individual behavior. Y₁, being the sum of two independent exponential random variables, follows a gamma distribution, which is a more flexible distribution than the exponential. Y₂, on the other hand, retains the same exponential distribution as the original observations. Another important aspect to consider is the independence of Y₁ and Y₂. In general, knowing that Y₁ is a sufficient statistic does not automatically imply that Y₁ and Y₂ are independent. To check for independence, we need to compare the joint p.d.f. of Y₁ and Y₂ with the product of their marginal p.d.f.s. If f(y₁, y₂; θ) = f(y₁; θ) * f(y₂; θ), then Y₁ and Y₂ are independent. In our case, we have: f(y₁, y₂; θ) = (1/θ²) e^(-y₁/θ), f(y₁; θ) = (1/θ²) e^(-y₁/θ) *y₁, f(y₂; θ) = (1/θ) e^(-y₂/θ). It is clear that f(y₁, y₂; θ) ≠ f(y₁; θ) * f(y₂; θ), so Y₁ and Y₂ are not independent. This is expected, as Y₂ is a component of Y₁, and therefore, they are related. The non-independence of Y₁ and Y₂ does not diminish the importance of Y₁ as a sufficient statistic. It simply means that the information contained in Y₂ is not independent of the information contained in Y₁. In summary, deriving the joint p.d.f. of Y₁ and Y₂ allows us to understand their relationship, derive their marginal distributions, and investigate their independence. This analysis provides valuable insights into the statistical properties of the sufficient statistic Y₁ and the auxiliary variable Y₂. Further analysis could involve using the joint p.d.f. to construct confidence intervals for θ or to perform hypothesis tests about θ.

In conclusion, this exploration has provided a comprehensive analysis of sufficient statistics and joint probability density functions (p.d.f.s) within the context of the exponential distribution. We began by introducing the fundamental concept of sufficient statistics, emphasizing their crucial role in simplifying statistical inference by capturing all the information relevant to the parameter of interest. We highlighted the factorization theorem as a powerful tool for identifying sufficient statistics and discussed their applications in parameter estimation and hypothesis testing. Next, we delved into the exponential distribution, a fundamental probability distribution widely used to model the time until an event occurs in a Poisson process. We examined its key properties, including the memoryless property and its relationship to the Poisson and gamma distributions. We also discussed parameter estimation and hypothesis testing procedures for the exponential distribution. The core of our analysis focused on a specific problem: finding the joint p.d.f. of the sufficient statistic Y₁ = X₁ + X₂ for θ and Y₂ = X₂, given a random sample X₁, X₂ from an exponential distribution with parameter θ. We meticulously derived the joint p.d.f. using the transformation formula, which involves the Jacobian determinant. We also determined the region of support for Y₁ and Y₂. The derived joint p.d.f. allowed us to gain valuable insights into the relationship between Y₁ and Y₂ and to confirm that Y₁ is indeed a sufficient statistic for θ. Furthermore, we derived the marginal distributions of Y₁ and Y₂, revealing that Y₁ follows a gamma distribution and Y₂ follows an exponential distribution. We also investigated the independence of Y₁ and Y₂, concluding that they are not independent, as expected. This comprehensive analysis demonstrates the power of sufficient statistics in simplifying statistical inference and the utility of joint p.d.f.s in understanding the relationships between random variables. The techniques and concepts discussed in this exploration are applicable to a wide range of statistical problems and provide a solid foundation for further study in statistical theory and applications. The ability to identify sufficient statistics and derive joint p.d.f.s is crucial for developing efficient estimators, constructing powerful hypothesis tests, and making informed decisions based on data. In summary, this exploration has provided a deep dive into the concepts of sufficient statistics and joint p.d.f.s, illustrating their importance in statistical inference and their application to the exponential distribution. The results and insights gained from this analysis can be valuable for researchers and practitioners working in various fields, including statistics, probability, and data science. By understanding and utilizing these concepts, we can unlock the full potential of statistical data and make more informed decisions.