Suppose that Y1, Y2, …, Yn denote a random sample of size n...

Suppose that $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample of size $n$ from a normal distribution with mean $\mu$ and variance $1 .$ Consider the first observation $Y_{1}$ as an estimator for $\mu$ a. Show that $Y_{1}$ is an unbiased estimator for $\mu$. b. Find $P\left(\left|Y_{1}-\mu\right| \leq 1\right)$ c. Look at the basic definition of consistency given in Definition 9.2. Based on the result of part (b), is $Y_{1}$ a consistent estimator for $\mu$ ?

Suppose that $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample of size $n$ from a normal distribution with mean
$\mu$ and variance $1 .$ Consider the first observation $Y_{1}$ as an estimator for $\mu$
a. Show that $Y_{1}$ is an unbiased estimator for $\mu$.
b. Find $P\left(\left|Y_{1}-\mu\right| \leq 1\right)$
c. Look at the basic definition of consistency given in Definition 9.2. Based on the result of part
(b), is $Y_{1}$ a consistent estimator for $\mu$ ?

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Key Concepts

Properties of the Normal Distribution

The normal distribution is a fundamental probability distribution characterized by its bell-shaped density function, symmetry about the mean, and its parameters. Its mathematical properties facilitate the calculation of probabilities and the derivation of statistical inferences, such as determining the likelihood that an observation falls within a given interval around the mean.

Unbiased Estimator

An estimator is said to be unbiased if the expected value of the estimator is equal to the parameter it is intended to estimate. This means that on average, the estimator neither overestimates nor underestimates the true parameter value, ensuring that the estimation process does not systematically favor any particular direction.

Consistency of Estimators

Consistency refers to the property of an estimator where, as the sample size increases, the estimator converges in probability to the true parameter value. In other words, given any small deviation from the true parameter, the probability that the estimator will be within that deviation from the parameter becomes arbitrarily high as the sample size tends to infinity.

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