If X1, X2 is a random sample of size 2 from a distribution...

If $X_{1}, X_{2}$ is a random sample of size 2 from a distribution having pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty, 0<\theta<\infty$, zero elsewhere, find the joint pdf of the sufficient statistic $Y_{1}=X_{1}+X_{2}$ for $\theta$ and $Y_{2}=X_{2}$. Show that $Y_{2}$ is an unbiased estimator of $\theta$ with variance $\theta^{2}$. Find $E\left(Y_{2} \mid y_{1}\right)=\varphi\left(y_{1}\right)$ and the variance of $\varphi\left(Y_{1}\right)$

Key Concepts

Conditional Expectation

Conditional expectation is the expected value of a random variable given that another related variable or event is known. It is used to find the best prediction of one variable based on the information from another. In this problem, calculating the conditional expectation E(Y2 | Y1 = y1) helps in understanding the relationship between the sufficient statistic and the original variable, thereby playing a crucial role in variance calculations and further inference.

Variance of an Estimator

The variance of an estimator measures the spread of its sampling distribution and thus reflects the precision of the estimator. Lower variance indicates more reliable estimates. Determining the variance of an unbiased estimator, as done in this problem, is essential for evaluating its efficiency and for constructing confidence intervals around the parameter estimate.

Unbiased Estimators

An estimator is unbiased if its expected value equals the true value of the parameter it is estimating. In statistical inference, finding an unbiased estimator is desirable because, on average, it correctly estimates the parameter. In this context, demonstrating that a statistic is an unbiased estimator involves showing that its expected value under the sampling distribution is equal to the parameter of interest.

Exponential Distribution

The exponential distribution is a continuous probability distribution characterized by its rate or scale parameter. It is typically used to model the time between events in a Poisson process. The probability density function (pdf) is defined for positive values and is notable for its memoryless property, which implies that the distribution of the future waiting time is independent of the elapsed time.

Transformation of Random Variables

Transforming random variables involves changing from one set of variables to another using a function, and it is a common method in probability to find the joint pdf of new statistics derived from sample data. When a transformation is applied, the joint pdf of the new variables is obtained by incorporating the Jacobian of the transformation. This approach is crucial in deriving the distributions of sufficient statistics or other functions of the sample data.

Sufficient Statistics

A sufficient statistic for a parameter is a function of the sample data that contains all the information needed to estimate the parameter. In the context of the exponential distribution, the sum of the sample data (for example, the sum of two independent exponential random variables) is a sufficient statistic for the scale parameter. This concept reduces the complexity in parameter estimation and inference because further details of the sample do not contribute additional information.

Please give Ace some feedback