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Background

The data at hand consists of $n$ iid random variables represented as $X_j$, where $j \in \{1,\ldots,n\}$. We know $\forall i,\, \operatorname{E}\left(X_i\right) = \mu$, and that $\operatorname{Var}\left(X_i\right) = \sigma^2$.

Suppose we generate $B$ bootstrap samples from this data, with the $i$th element of the $b$th bootstrap sample denoted by $X_i^{∗b}$.

Question

What are $\operatorname{E}\left(X_i^{∗b}\right)$ and $\operatorname{Var}\left(X_i^{∗b}\right)$?

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1 Answer 1

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(i) Showing that $\operatorname{E}\left(X_i^{∗b}\right) = \mu$

$X_i^{∗b}$ is "composed" of random variables. To simplify the process of finding $\operatorname{E}\left(X_i^{∗b}\right)$, start with a conditional expectation.

\begin{equation*} \operatorname{E}\left(X_i^{∗b}\,\vert\,X_1,\ldots,X_n\right) = \sum_{j = 1}^{n}\frac{X_j}{n} \end{equation*} This expected value is a random variable itself. Therefore: \begin{equation*} \begin{split} \operatorname{E}\left(X_i^{∗b}\right) &= \operatorname{E}\left[\operatorname{E}\left(X_i^{∗b}\,\vert\,X_1,\ldots, X_n\right)\right]\\ &= \operatorname{E}\left(\sum_{j = 1}^{n}\frac{X_j}{n}\right)\\ & = \frac{1}{n}\sum_{j = 1}^{n}\operatorname{E}\left(X_j\right)\\ & = \frac{1}{n}\sum_{j = 1}^{n}\mu\\ & = \mu.\,\square \end{split} \end{equation*}

(ii) Showing that $\operatorname{Var}\left(X_i^{∗b}\right) = \sigma^2$

Recall that: \begin{equation*} \begin{split} \operatorname{Var}\left(X_i^{∗b}\right) &= \operatorname{E}\left[\left(X_i^{∗b}\right)^2\right] - \left[\operatorname{E}\left(X_i^{∗b}\right)\right]^2\\ &= \operatorname{E}\left[\left(X_i^{∗b}\right)^2\right] - \mu^2. \end{split} \end{equation*} Let us find $\operatorname{E}\left[\left(X_i^{∗b}\right)^2\right]$ in a manner similar to (i). We will need to also apply the ``law of the unconscious statistician'', which states that: \begin{equation*} \operatorname{E}[g(X)] = \sum _{x}g(x)f_{X}(x) \end{equation*} for some function $g$ acting on a random variable $X$. \begin{equation*} \operatorname{E}\left[\left(X_i^{∗b}\right)^2\,\bigg|\,X_1,\ldots,X_n\right] = \sum_{j = 1}^{n}\frac{\left(X_j\right)^2}{n} \end{equation*} \begin{equation*} \begin{split} \implies \operatorname{E}\left[\left(X_i^{∗b}\right)^2\right] &= \operatorname{E}\left\{\operatorname{E}\left[\left(X_i^{∗b}\right)^2\,\bigg|\, X_1,\ldots,X_n\right]\right\}\\ &= \operatorname{E}\left[\sum_{j = 1}^{n}\frac{\left(X_j\right)^2}{n}\right]\\ & = \frac{1}{n}\sum_{j = 1}^{n}\operatorname{E}\left[\left(X_j\right)^2\right]\\ & = \frac{1}{n}\sum_{j = 1}^{n}\left(\sigma^2 + \mu^2\right)\\ & = \sigma^2 + \mu^2 \end{split} \end{equation*} \begin{equation*} \begin{split} \implies \operatorname{Var}\left(X_i^{∗b}\right) &= \sigma^2 + \mu^2 - \mu^2\\ &= \sigma^2.\,\square \end{split} \end{equation*}

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