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
−∞
Variance:
∞ ∞
𝜎 2 = 𝑉(𝑋) = ∫ (𝑥 − 𝜇)2 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑥 2 𝑓(𝑥) 𝑑𝑥 − 𝜇2
−∞ −∞
3
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
−∞
Variance:
∞ ∞
𝜎 2 = 𝑉(𝑋) = ∫ (𝑥 − 𝜇)2 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑥 2 𝑓(𝑥) 𝑑𝑥 − 𝜇2
−∞ −∞
3
