This document provides a summary of all theory and formulas from the chapters 4 to 12 inclusive (except for chapter 6) book 'Applied Statistics and Probability for Engineers' taught in the statistic course 2DD80.
Summary statistics for IE
All content and pictures are from the following book:
Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for
Engineers: Study Guide.
Contents
Chapter 4: Continuous random variables and probability distributions .................................. 3
4.1: Probability distributions and probability density functions ............................................ 3
4.2: Cumulative distribution functions ................................................................................ 3
4.3: Mean and variance of a continuous random variable .................................................. 3
4.4: Continuous uniform distribution .................................................................................. 4
4.5: Normal distribution ...................................................................................................... 4
4.6: Normal approximation to the binomial and Poisson distributions ................................ 4
4.7: Exponential distribution............................................................................................... 5
4.8: Erlang and gamma distributions ................................................................................. 5
4.9: Weibull distribution ..................................................................................................... 6
4.10: lognormal distribution................................................................................................ 6
4.11: Beta distribution ........................................................................................................ 6
Chapter 5: joint probability distributions ................................................................................. 7
5.1: joint probability distributions for two random variables ................................................ 7
5.2: Conditional probability distributions and independence............................................... 8
5.3: Joint probability distributions for more than two random variables .............................. 8
5.4: Covariance and correlation ......................................................................................... 9
5.5: Common joint distributions ....................................................................................... 10
5.6: Linear functions of random variables ........................................................................ 11
5.7: General functions of random variables ..................................................................... 11
Chapter 7: point estimation of parameters and sampling distributions ................................. 11
7.1: point estimation ........................................................................................................ 11
7.2: Sampling distributions and the central limit theorem ................................................. 12
7.3: General concepts of point estimation ........................................................................ 12
Chapter 8: statistical intervals for a single sample ............................................................... 13
8.1: confidence interval on the mean of a normal distribution, variance known ................ 13
8.2: confidence interval on the mean f a normal distribution, variance unknown .............. 14
8.3: confidence interval on the variance and standard deviation of normal distribution .... 14
1
, 8.4: large sample confidence interval for a population proportion .................................... 14
8.7: Tolerance and prediction intervals ............................................................................ 15
Chapter 9: tests of hypotheses for a single sample ............................................................. 15
9.1: hypothesis testing ..................................................................................................... 15
9.2: tests on the mean of a normal distribution, variance known ...................................... 15
9.3: tests on the mean of a normal distribution, variance unknown .................................. 17
9.4: tests on variance and standard deviation of normal distribution ................................ 17
9.5: tests on population proportion................................................................................... 17
9.7: goodness of fit .......................................................................................................... 18
9.8: Contingency table tests ............................................................................................ 19
Chapter 10: Statistical inference for two samples ................................................................ 19
10.1: inference on difference in means of two normal distributions, variances known ...... 19
10.2: Hypotheses test on the difference in means, variances unknown ........................... 20
10.4: paired t-Test ........................................................................................................... 22
10.6: inference on two population proportions ................................................................. 23
Chapter 11: simple linear regression and correlation........................................................... 24
11.1: empirical models..................................................................................................... 24
11.2: simple linear regression .......................................................................................... 24
11.3: properties of the least squares estimators .............................................................. 26
11.4: Hypothesis tests in simple linear regression ........................................................... 26
11.5: confidence intervals ................................................................................................ 26
11.6: prediction of new observations ............................................................................... 27
11.8: Correlation .............................................................................................................. 27
11.9: Regression on transformed variables ..................................................................... 27
Chapter 12: multiple linear regression ................................................................................. 27
12.1: multiple linear regression model ............................................................................. 27
12.2: hypothesis tests in multiple linear regression .......................................................... 28
12.3: confidence intervals in multiple linear regression .................................................... 28
12.5: model adequacy checking ...................................................................................... 29
2
, Chapter 4: Continuous random variables and probability distributions
4.1: Probability distributions and probability density functions
Continuous random variable: random variable with an interval (either finite or infinite) of
real number for its range.
Probability density function (f(x)): can be used to describe probability distribution of
continuous random variable X. If an interval is likely to contain a value for X, its probability is
large and it corresponds to large values for f(x).
3 conditions:
1. 𝑓(𝑥) ≥ 0
∞
2. ∫−∞ 𝑓(𝑥) 𝑑𝑥 = 1
𝑏
3. 𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) = ∫𝑎 𝑓(𝑥) 𝑑𝑥 = area under f(x) from a to b for any a and b
Histogram: approximation to a probability density function. The area of each interval of the
histogram equals the relative frequency (proportion) of measurements in the interval.
𝑃(𝑋 = 𝑥) = 0
If X is a continuous random variable, for any x1 and x2:
𝑃(𝑥1 ≤ 𝑋 ≤ 𝑥2 ) = 𝑃(𝑥1 < 𝑋 ≤ 𝑥2 ) = 𝑃(𝑥1 ≤ 𝑋 < 𝑥2 ) = 𝑃(𝑥1 < 𝑋 < 𝑥2 )
4.2: Cumulative distribution functions
The cumulative distribution function of a continuous random variable X is:
𝑥
𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫−∞ 𝑓(𝑢) 𝑑𝑢 𝑓𝑜𝑟 − ∞ < 𝑥 < ∞
Any < can be changed into ≤ in the definition of F(x). The probability density function of a
continuous random variable can be determined from the cumulative distribution function by
differentiating.
𝑑 𝑥 𝑑𝐹(𝑥)
∫ 𝑓(𝑢) 𝑑𝑢
𝑑𝑥 −∞
= 𝑓(𝑥) and 𝑓(𝑥) =
𝑑𝑥
4.3: Mean and variance of a continuous random variable
Suppose that X is a continuous random variable with probability density function f(x).
Mean / expected value:
∞
𝜇 = 𝐸(𝑋) = ∫ 𝑥𝑓(𝑥) 𝑑𝑥
−∞
∞
𝐸[ℎ(𝑋)] = ∫ ℎ(𝑥)𝑓(𝑥) 𝑑𝑥
−∞
Voordelen van het kopen van samenvattingen bij Stuvia op een rij:
Verzekerd van kwaliteit door reviews
Stuvia-klanten hebben meer dan 700.000 samenvattingen beoordeeld. Zo weet je zeker dat je de beste documenten koopt!
Snel en makkelijk kopen
Je betaalt supersnel en eenmalig met iDeal, creditcard of Stuvia-tegoed voor de samenvatting. Zonder lidmaatschap.
Focus op de essentie
Samenvattingen worden geschreven voor en door anderen. Daarom zijn de samenvattingen altijd betrouwbaar en actueel. Zo kom je snel tot de kern!
Veelgestelde vragen
Wat krijg ik als ik dit document koop?
Je krijgt een PDF, die direct beschikbaar is na je aankoop. Het gekochte document is altijd, overal en oneindig toegankelijk via je profiel.
Tevredenheidsgarantie: hoe werkt dat?
Onze tevredenheidsgarantie zorgt ervoor dat je altijd een studiedocument vindt dat goed bij je past. Je vult een formulier in en onze klantenservice regelt de rest.
Van wie koop ik deze samenvatting?
Stuvia is een marktplaats, je koop dit document dus niet van ons, maar van verkoper evamaria45. Stuvia faciliteert de betaling aan de verkoper.
Zit ik meteen vast aan een abonnement?
Nee, je koopt alleen deze samenvatting voor €10,49. Je zit daarna nergens aan vast.
