Problem 1.23: Show that

\(\left( \begin{matrix}

n \\

{{n}_{1}},{{n}_{2}},{{n}_{3}},....,{{n}_{k}} \\

\end{matrix} \right)=\left( \begin{matrix}

n-1 \\

{{n}_{1}}-1,{{n}_{2}},{{n}_{3}},....,{{n}_{k}} \\

\end{matrix} \right)+\left( \begin{matrix}

n-1 \\

{{n}_{1}},{{n}_{2}}-1,{{n}_{3}},....,{{n}_{k}} \\

\end{matrix} \right)+...+\left( \begin{matrix}

n-1 \\

{{n}_{1}},{{n}_{2}},{{n}_{3}},....,{{n}_{k}}-1 \\

\end{matrix} \right)\)

Problem 1.19:

1. Compute
   \(\left( \begin{matrix}
   \frac{1}{2} \\
   4 \\
   \end{matrix} \right)\)
   and
   \(\left( \begin{matrix}
   -3 \\
   3 \\
   \end{matrix} \right)\)
2. Approximate \(\sqrt{5}\) using Stirling’s approximation.

Problem 5.1: If X has a discrete uniform distribution \(f(x)=\frac{1}{k}\), for \(x=1,2...,k\), show that

1. The mean is \(\mu =\frac{k+1}{2}\)
2. Its variance is \({{\sigma }^{2}}=\frac{{{k}^{2}}-1}{12}\).

Problem 5.2 If X has a discrete uniform distribution \(f(x)=\frac{1}{k}\), for \(x=1,2...,k\), show that its moment generating function is given by:

Problem 5.6

1. Consider the probability function defined by
   \[f(x)=\left\{ \begin{aligned}
   & \theta \text{ if }x=1 \\
   & 1-\theta \text{ if }x=0 \\
   \end{aligned} \right.\]
   Compute \(E\left( X \right)\) and \(\operatorname{var}\left( X \right)\) .
2. Define \(Y=\sum\limits_{i=1}^{n}{{{X}_{i}}}\), compute \(E\left( Y \right)\) and \(\operatorname{var}\left( Y \right)\) .

Problem 5.14:

1. Prove that
   \(M_{X-\mu }^{(k)}(t)=E\left( {{(X-\mu )}^{k}}{{e}^{t(X-\mu )}} \right)\)
2. For a binomial distribution with parameters n and p, prove that its variance is

\(np(1-p)\)

Problem 5.20: For a probability function defined by

\(p(n)=\theta {{(1-\theta )}^{n-1}}\)

Compute the moment generating function.

Problem 5.23: Prove the "memoryless property" of the geometric distribution. This is, prove that

\(\Pr (X=x+n|X>n)=\Pr (X=x)\)

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Solution: The downloadable solution consists of 9 pages, 281 words.
Deliverable: Word Document