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17\fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
18\fancyhead[CO,CE]{Andrew Lorimer}
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21
22\begin{document}
23
24 \title{Statistics}
25 \author{}
26 \date{}
27 \maketitle
28
29 \section{Linear combinations of random variables}
30
31 \subsection*{Continuous random variables}
32
33 A continuous random variable \(X\) has a pdf \(f\) such that:
34
35 \begin{enumerate}
36 \item \(f(x) \ge 0 \forall x \)
37 \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
38 \end{enumerate}
39
40 \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
41
42 \subsubsection*{Linear functions \(X \rightarrow aX+b\)}
43
44 \begin{align*}
45 \Pr(Y \le y) &= \Pr(aX+b \le y) \\
46 &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
47 &= \int^{\dfrac{y-b}{a}}_{-\infty} f(x) \> dx
48 \end{align*}
49
50 \begin{align*}
51 \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
52 \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
53 \end{align*}
54
55 \subsection*{Linear combination of two random variables}
56
57 \begin{align*}
58 \textbf{Mean:} && \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
59 \textbf{Variance:} && \operatorname{Var}(aX+bY) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \tag{if \(X\) and \(Y\) are independent}\\
60 \end{align*}
61
62 \section{Sample mean}
63
64 \[ \overline{x} = \dfrac{\Sigma x}{n} \]
65
66 where \(n\) is the size of the sample (number of sample points)
67
68 \subsubsection*{\colorbox{cas}{On CAS:}}
69
70 \begin{enumerate}
71 \item Spreadsheet
72 \item In cell A1: \verb;mean(randNorm(sd, mean, sample size));
73 \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
74 \item Input range as A1:An where \(n\) is the number of samples
75 \item Graph \(\rightarrow\) Histogram
76 \end{enumerate}
77
78 \subsubsection*{Sample size of \(n\)}
79
80 \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
81
82 Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\)
83
84
85
86\end{document}