Normal distribution

Definition: $X\sim N(\mu ,{\sigma }^2)$

• Continuous random variable
• based on Standard normal distribution
• We say that $X$ is normal random variable, or simply that $X$ is normally distributed, with parameters $\mu$ and ${\sigma }^2$ denoted by $X\sim N(\mu ,{\sigma }^2)$
  • $X\sim N(5,10)$
• Properties:
  • ${\int }_{-\infty }^{\infty }f(x)dx=1$
  • Expected value $E[X]=\mu$
  • Variance $Var(x)={\sigma }^2$
  • Graph of $f$ is symmetric in the line $x=\mu$
  • $f$ is maximized when $x=\mu$
  • $f$ has two Infection points at $x=\mu \pm \mu$
• Conditions
  • $f(x)\ge 0\forall x\in (-\infty ,\infty )$

Probability density function

• $f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}},\infty <x<\infty$

Normal approximation to Binomial distribution:

• Use The DeMoivre-Laplace limit theorem to evaluate a Binomial distribution when the number of trials become large.
• $X\sim B(n,p)\to X\sim N(np,npq)$ where:
  • $np\ge 5$
  • $nq=n(1-p)\ge 5$
• Continuity correction:
  • Convert from discrete integer valued (binomial) to continuous (normal)
  • $P(X=i)$ as $P(i-0.5<X<i+0.5)$
  • $P(X<i)\to P(X<i+0.5)$
  • $P(X\le i)\to P(X<-0.5)$
  • $P(X>i)\to P(X<i+0.5)$
  • $P(X\ge i)\to P(X<i-0.5)$

Sum of independent normal distribution:

• Sum of independent variables
• If $X_i\sim N({\mu }_i,{\sigma }_i^2)$, then $\sum _{i=1}^n{X}_i\sim N(\sum _{i=1}^n{\mu }_i,\sum _{i=1}^n{\sigma }_i^2)$

Testing Normality of sample:

• If the history for a variable $X_i$ in a multivariate appears reasonably symmatric, we can check further by counting the number of observation in certain intervals (quantile) to see if it follows the normal distribution
• We can use Q-Q Plot to assess the assumption of normality
  • with plots the sample quatile versus the expected quantile if the observations were normally distributed
  • If the points form a straight line with high Correlation Coefficient, it means that the assumption that sample collected is normal is non-rejectable
  • The idea is to plot the data and its z-score, it the sample is normally distributed, their z-score should increase linearly (z-score is already standardized by $(x-\mu )/\sigma$
• If we have $n$ elements observed and order them, then quantile ($q_j$) of $j$th element$=p_{(j)}=\frac{j-0.5}{n}=P[Z\le q_{(j)}]$
  • $\to q_{(j)}=\varphi (p_{(j)})=\varphi \left(\frac{j-0.5}{n}\right)$
• Steps:
  • $H_0$: sample normally distributed, $H_1$…
  • for each data, find $q_{(j)}$
  • then plot, $(q_{(j)},x_{(j)})$ where $x_{(j)}$ is the corresponding data point
  • Find the Correlation Coefficient of the line, it is our test statistics
  • Compare it against Chi-squared distribution with n dof, reject if $ts<{\chi }_n^2(\alpha )$

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