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CME 106 - Introduction to Probability and Statistics for Engineers

By Afshine Amidi and Shervine Amidi Statistics Cheat Sheet

Parameter estimation

Definitions

Random sample A random sample is a collection of $n$ random variables $X_1, .., X_n$ that are independent and identically distributed with $X$.

Estimator An estimator is a function of the data that is used to infer the value of an unknown parameter in a statistical model.

Bias The bias of an estimator $hat{theta}$ is defined as being the difference between the expected value of the distribution of $hat{theta}$ and the true value, i.e.:

[boxed{textrm{Bias}(hat{theta})=E[hat{theta}]-theta}]

Statistics formulas cheat sheet step by step

Remark: an estimator is said to be unbiased when we have $E[hat{theta}]=theta$.

Estimating the mean

Sample mean The sample mean of a random sample is used to estimate the true mean $mu$ of a distribution, is often noted $overline{X}$ and is defined as follows:

[boxed{overline{X}=frac{1}{n}sum_{i=1}^nX_i}]

Remark: the sample mean is unbiased, i.e $E[overline{X}]=mu$.

Characteristic function for sample mean The characteristic function for a sample mean is noted $psi_{overline{X}}$ and is such that:

[boxed{psi_{overline{X}}(omega)=psi_{X}^nleft(frac{omega}{n}right)}]

Central Limit Theorem Let us have a random sample $X_1, .., X_n$ following a given distribution with mean $mu$ and variance $sigma^2$, then we have:

[boxed{overline{X}underset{nrightarrow+infty}{sim}mathcal{N}left(mu, frac{sigma}{sqrt{n}}right)}]

Estimating the variance

Sample variance The sample variance of a random sample is used to estimate the true variance $sigma^2$ of a distribution, is often noted $s^2$ or $hat{sigma}^2$ and is defined as follows:

[boxed{s^2=hat{sigma}^2=frac{1}{n-1}sum_{i=1}^n(X_i-overline{X})^2}]

Remark: the sample variance is unbiased, i.e $E[s^2]=sigma^2$. Install mac os x virtualbox dmg file.

Chi-Squared relation with sample variance Printmaster platinum for mac. Let $s^2$ be the sample variance of a random sample. We have:

[boxed{frac{s^2(n-1)}{sigma^2}simchi_{n-1}^2}]

Confidence intervals

Definitions

Confidence level A confidence interval with confidence level $1-alpha$ is such that $1-alpha$ of the time, the true value is contained in the confidence interval.

Confidence interval A confidence interval $CI_{1-alpha}$ with confidence level $1-alpha$ of a true parameter $theta$ is such that:

With the notation of the example above, a possible $1-alpha$ confidence interval for $theta$ is given by $CI_{1-alpha} = [x_1, x_2]$.

Confidence interval for the mean

When determining a confidence interval for the mean $mu$, different test statistics have to be computed depending on which case we are in. The table below sums it up.

Distribution of $X_i$	Sample size $n$	Variance $sigma^2$	Statistic	$small 1-alpha$ confidence interval
$X_isimmathcal{N}(mu, sigma)$	any	known	$displaystylefrac{overline{X}-mu}{frac{sigma}{sqrt{n}}}simmathcal{N}(0,1)$	$left[overline{X}-z_{frac{alpha}{2}}frac{sigma}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{sigma}{sqrt{n}}right]$
$X_isim$ any distribution	large	known	$displaystylefrac{overline{X}-mu}{frac{sigma}{sqrt{n}}}simmathcal{N}(0,1)$	$left[overline{X}-z_{frac{alpha}{2}}frac{sigma}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{sigma}{sqrt{n}}right]$
$X_isim$ any distribution	large	unknown	$displaystylefrac{overline{X}-mu}{frac{s}{sqrt{n}}}simmathcal{N}(0,1)$	$left[overline{X}-z_{frac{alpha}{2}}frac{s}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{s}{sqrt{n}}right]$
$X_isimmathcal{N}(mu, sigma)$	small	unknown	$displaystylefrac{overline{X}-mu}{frac{s}{sqrt{n}}}sim t_{n-1}$	$left[overline{X}-t_{frac{alpha}{2}}frac{s}{sqrt{n}},overline{X}+t_{frac{alpha}{2}}frac{s}{sqrt{n}}right]$
$X_isim$ any distribution	small	known or unknown	Go home!	Go home!

Note: a step by step guide to estimate the mean, in the case when the variance in known, is detailed here.

Confidence interval for the variance

The single-line table below sums up the test statistic to compute when determining the confidence interval for the variance.

Distribution of $X_i$	Sample size $n$	Mean $mu$	Statistic	$small 1-alpha$ confidence interval
$X_isimmathcal{N}(mu,sigma)$	any	known or unknown	$displaystylefrac{s^2(n-1)}{sigma^2}simchi_{n-1}^2$	$left[frac{s^2(n-1)}{chi_2^2},frac{s^2(n-1)}{chi_1^2}right]$

Note: a step by step guide to estimate the variance is detailed here.

Hypothesis testing

Statistics Formulas Cheat Sheet Step By Step

General definitions

Type I error In a hypothesis test, the type I error, often noted $alpha$ and also called 'false alarm' or significance level, is the probability of rejecting the null hypothesis while the null hypothesis is true. If we note $T$ the test statistic and $R$ the rejection region, then we have:

[boxed{alpha=P(Tin R|H_0{small