Question

B7. (a) Derive the moment generating function of the Poisson distribution with parameter A. [4 marks] (b) Let $\{X_i, i = 1, \dots, n\}$ be independent random variables given by the Bernoulli distribution with probability of success $p \in (0, 1)$, that is $P(X_i = 1) = p$, $P(X_i = 0) = 1 - p$. Let $S_n = X_1 + X_2 + \dots + X_n\ for \(n \ge 1$. Derive the moment generating function $M_{S_n}$ for $S_n$. [4 marks] (c) Given that $\lim_{x \to 0} (1 + \alpha x)^{1/x} = e^\alpha$, derive conditions on $p$ (depending on $n$) and on $n$ itself that would imply $M_{S_n} \to M_P$ pointwise where $M_P$ is the moment generating function of a Poisson random variable. [2 marks] (d) State the central limit theorem. [3 marks] (e) Interpret the central limit theorem. [3 marks] (f) Given that $S_n \sim Bin(n, p)$ for all $n \ge 1$, show that the central limit theorem predicts that for large $n$ we have $Bin(n, p) \approx N(np, np(1 - p))$. [4 marks]

          B7.
(a) Derive the moment generating function of the Poisson distribution with parameter A.
[4 marks]
(b) Let \(\{X_i, i = 1, \dots, n\}\) be independent random variables given by the Bernoulli distribution with
probability of success \(p \in (0, 1)\), that is
\(P(X_i = 1) = p\), \(P(X_i = 0) = 1 - p\).
Let
\(S_n = X_1 + X_2 + \dots + X_n\
for \(n \ge 1\). Derive the moment generating function \(M_{S_n}\) for \(S_n\).
[4 marks]
(c) Given that \(\lim_{x \to 0} (1 + \alpha x)^{1/x} = e^\alpha\), derive conditions on \(p\) (depending on \(n\)) and on \(n\) itself that
would imply \(M_{S_n} \to M_P\) pointwise where \(M_P\) is the moment generating function of a Poisson
random variable.
[2 marks]
(d) State the central limit theorem.
[3 marks]
(e) Interpret the central limit theorem.
[3 marks]
(f) Given that \(S_n \sim Bin(n, p)\) for all \(n \ge 1\), show that the central limit theorem predicts that for large
\(n\) we have \(Bin(n, p) \approx N(np, np(1 - p))\).
[4 marks]

$B7. (a) Derive the moment generating function of the Poisson distribution with parameter A. [4 marks] (b) Let {Xi, i = 1, …, n} be independent random variables given by the Bernoulli distribution with probability of success p ∈ (0, 1), that is P(Xi = 1) = p, P(Xi = 0) = 1 - p. Let Sn = X1 + X2 + … + Xnfor n ≥ 1. Derive the moment generating function MSn for Sn. [4 marks] (c) Given that limx → 0 (1 + α x)^1/x = e^α, derive conditions on p (depending on n) and on n itself that would imply MSn→ MP pointwise where MP is the moment generating function of a Poisson random variable. [2 marks] (d) State the central limit theorem. [3 marks] (e) Interpret the central limit theorem. [3 marks] (f) Given that Sn ∼ Bin(n, p) for all n ≥ 1, show that the central limit theorem predicts that for large n we have Bin(n, p) ≈ N(np, np(1 - p)). [4 marks]$

Added by Brian G.

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