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Workshop solutions for Math 103 Probability: Week 13

1.
1. (a)
  
  A random variable $R$ is a function $R:\Omega\rightarrow\mathbb{R}$ .
2. (b)
  
  The induced sample space ${\mathcal{S}}$ is the range of values taken by the random variable $R$ defined on $\Omega$ , that is ${\mathcal{S}}=\{R(\omega):\omega\in\Omega\}$ .
3. (c)
  
  The probability mass function of a discrete random variable, $R$ , is defined by
  
  $\displaystyle p_{R}(r)={\rm P}(R=r)\mbox{ for }r=0,1,2,\dots.$
4. (d)
  
  The cumulative distribution function of a random variable $R$ is a function $F:\mathbb{N}\rightarrow\mathbb{R}$ given by
  
  $\displaystyle F(m)={\rm P}(R\leq m)=\sum^{m}_{r=0}p_{R}(r).$
5. (e)
  
  The expected value of a discrete random variable $R$ is
  
  $\displaystyle{\rm E}[R]=\sum_{r=0}^{\infty}rp_{R}(r).$
6. (f)
  
  The variance of a random variable $R$ is
  
  $\displaystyle{\rm Var}(R)={\rm E}[(R-{\rm E}[R])^{2}].$
7. (g)
  
  The standard deviation of $R$ is the square root of the variance.
2.
1. (a)
  
  $\displaystyle{\rm P}(R>2)=\sum_{r=3}^{8}p_{R}(r)=\frac{6}{8}=\frac{3}{4}$
2. (b)
  
  $F(r)={\rm P}(R\leq r)=\sum_{s=0}^{r}p_{R}(r)=\left\{\begin{array}[]{rl}0&\mbox% { for }r\leq 0\\ \frac{r}{8}&\mbox{ for }1\leq r\leq 8\\ 1&\mbox{ for }r\geq 8.\end{array}\right.$

3.

	$\displaystyle{\rm E}[g(R)+h(R)]$	$\displaystyle=$	$\displaystyle\sum_{r=0}^{\infty}[g(r)+h(r)]\,p_{R}(r)\quad\mbox{def }{\rm E}$
		$\displaystyle=$	$\displaystyle\sum_{r=0}^{\infty}[g(r)p_{R}(r)+h(r)p_{R}(r)]$
		$\displaystyle=$	$\displaystyle\sum_{r=0}^{\infty}g(r)p_{R}(r)+\sum_{r=0}^{\infty}h(r)p_{R}(r)% \quad\mbox{lin }\Sigma$
		$\displaystyle=$	$\displaystyle{\rm E}[g(r)]+{\rm E}[h(r)],\quad\mbox{def }{\rm E}$

	$\displaystyle{\rm E}[cg(r)]$	$\displaystyle=$	$\displaystyle\sum_{r=0}^{\infty}cg(r)p_{R}(r)\quad\mbox{def }{\rm E}$
		$\displaystyle=$	$\displaystyle c\sum_{r=0}^{\infty}g(r)p_{R}(r)\quad\mbox{common factor}$
		$\displaystyle=$	$\displaystyle=c\,{\rm E}[g(r)],$

4.

$\displaystyle{\rm P}(X=x)$ $\displaystyle=$ $\displaystyle{20\choose x}(0.2)^{x}(0.8)^{20-x}\mbox{ for }x=0,1,\dots,20$

$\displaystyle{\rm E}(X)$ $\displaystyle=$ $\displaystyle 20\times 0.2=4,$

$\displaystyle{\rm P}(X>2)$ $\displaystyle=$ $\displaystyle 1-{\rm P}(X=0)-{\rm P}(X=1)-{\rm P}(X=2)$

$\displaystyle=$ $\displaystyle 0.794.$

1-pbinom(2,size=20,prob=0.2)
5.

$\displaystyle{\rm P}(R\geq n)$ $\displaystyle=$ $\displaystyle\sum_{r=n}^{\infty}p_{R}(r)$

$\displaystyle=$ $\displaystyle\sum_{r=n}^{\infty}(1-\theta)^{r}\theta$

$\displaystyle=$ $\displaystyle\theta(1-\theta)^{n}\sum_{r=0}^{\infty}(1-\theta)^{r}$

$\displaystyle=$ $\displaystyle\theta(1-\theta)^{n}\frac{1}{\theta}$

$\displaystyle=$ $\displaystyle(1-\theta)^{n}.$

$\displaystyle{\rm P}(R=n+r|R\geq n)$ $\displaystyle=$ $\displaystyle\frac{{\rm P}(R=n+r\mbox{ and }R\geq n)}{{\rm P}(R\geq n)}$

$\displaystyle=$ $\displaystyle\frac{{\rm P}(R=n+r)}{{\rm P}(R\geq n)}$

$\displaystyle=$ $\displaystyle\frac{(1-\theta)^{n+r}\theta}{(1-\theta)^{n}}$

$\displaystyle=$ $\displaystyle(1-\theta)^{r}\theta$

$\displaystyle=$ $\displaystyle p_{R}(r)={\rm P}(R=r).$

Lack of memory in independent (biased) coin tosses, i.e. the information that you’ve had $n$ tails previously tells you no more information about how many further coin tosses you need to wait before you see a head.