MatheMaticals Statistics 1st take home exam

For each \(n \geq 1\), let \(X_n\) be a random variable with the Binomial \((n,p_n)\) distribution. Suppose that \(p_n \rightarrow 0\) and \(n{p_n} \ rightarrow \inf\) as \(n \rightarrow \infty\) for sime \(0< \lambda< \infty\)

Problem 1. Show that there exist \(\{ {{\pi}_k} \}_{k \geq 0}\)such that

$\sum_{k=0}^{\infty}|p_{n,k}-\pi_k|^2 \rightarrow 0\ as\ n \rightarrow \infty $,

where $p_{n,k} \equiv P(X_n = k),\ k \geq 0,\ n \geq 1 $.

Under Assumption, we can use Binomial and Poisson's mgfs link.

mgf of \(X_n\) is computed as

\(M_{X_n}(t) = E[e^{tX_n}]= \sum_{x_n}^ne^{tx_n}\dbinom{n}{x_n}p_n^{x_n}(1-p_n)^{n-x_n}\)

Let \(x_n = k\),

\(M_{X_n}(t)=\sum_{x_n}^ne^{tk}\dbinom{n}{k}p_n^{k}(1-p_n)^{n-k}\)

\(=\sum_{k=0}^n\dbinom{n}{k}(p_ne^t)^k(1-p_n)^{n-k}= (p_ne^t+(1-p_n))^n,\ ((a+b)^n=\sum_{k}^n\dbinom{n}{k}a^{k}b^{n-k} )\)

Mgf of \(Poisson(\lambda)\) is expressed like \(M_Y(t)=e^{\lambda(e^t-1)}\)

Because \(p_n \rightarrow 0\) and \(n{p_n} \rightarrow \inf\) as \(n \rightarrow \infty\) for sime \(0< \lambda< \infty\),

\(lim_{n\rightarrow \infty}M_{X_n}(t)=lim_{n\rightarrow \infty}(p_ne^t+(1-p_n))^n= lim_{n\rightarrow \infty}(1+\frac{np(e^t-1)}{n})^n= e^{\lambda(e^t-1)}\)

Then \(lim_{n\rightarrow \infty}E[e^{tk}]= lim_{n\rightarrow \infty}\sum_{k}^ne^{tk}p_{n,k}\).

So, $ lim_{n \rightarrow \infty}(\sum_{k=0}^n({e^{tk}}p_{n,k}- \frac{e^{\lambda(e^t-1)}}{(n+1)e^{tk}}))=0$

\(\sum_{k=0}^\infty|p_{n,k}-{\pi}_k|^2 \rightarrow 0\ as\ n \rightarrow \infty\)

\(morbidity_{ij} = \beta_0 + \beta_1(low\ temperature_{ij})+\beta_2(low\ temperature_{ij} - \psi)_{+} + \epsilon_{ij}\)

\(where,\ (high\ temperature_{ij} - \psi)_{+}= (high\ temperature_{ij}-\psi)\times I(high\ temperature_{ij} > \psi), \epsilon_{ij} \sim N(0,\sigma^2)\)

\(mortality\)

\(\psi_{heat,i}=\beta_0 + \beta_1(high\ temperature_i)+ \beta_2(max\ humidity_i)+ \mathbf{\beta_3}ses_i^T + \epsilon_i\)

\(where\ \epsilon_i \sim N(0,\sigma^2),\ ses\ is\ Socioeconomic\ Status\)