Question

(2pt) Consider a population of size $N$ and simple random sampling (without replacement) with sample size $n$. Assume that the population takes values \{$x_1, \dots, x_N\}$ where $x_i \in \{0, 1\}$ and that the proportion of 1's satisfies $\frac{1}{N} \sum_{i=1}^{N} x_i \to p$ as $N \to \infty$ and $p \in (0, 1)$. Let $Y_n := n\overline{X}_n$ and prove that: $Y_n \stackrel{d}{\to} Y$ as $N \to \infty$ where $Y \sim B(n, p)$.

          (2pt) Consider a population of size $N$ and simple random sampling (without replacement) with sample size $n$. Assume that the population takes values \{$x_1, \dots, x_N\}$ where $x_i \in \{0, 1\}$ and that the proportion of 1's satisfies $\frac{1}{N} \sum_{i=1}^{N} x_i \to p$ as $N \to \infty$ and $p \in (0, 1)$. Let $Y_n := n\overline{X}_n$ and prove that:

$Y_n \stackrel{d}{\to} Y$
as $N \to \infty$ where $Y \sim B(n, p)$.

$(2pt) Consider a population of size N and simple random sampling (without replacement) with sample size n. Assume that the population takes values {x1, …, xN} where xi ∈{0, 1} and that the proportion of 1's satisfies (1)/(N)∑i=1^N xi → p as N →∞ and p ∈ (0, 1). Let Yn := nXn and prove that: Yn d→ Y as N →∞ where Y ∼ B(n, p).$

Added by Julie G.

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