Completed notes of the course

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1 Random variables • RANDOM VARIABLE: a variable characterized by a single (different) numerical value associated to each outcome of an experiment (or a measurement) • ➔ random variables are stochastic variables described by a statistic distribution o Random variables can be of two different types: ▪ Continuous (ex. electric power, length, pressure, temperature, weight) ▪ Discrete (ex. number of scratches on a surface, number of nonconforming parts in a sample) • PROPERTIES: given x as a random variable: o R is the domain of X → P (X ϵ R) = 1 o The probability that x belongs to any subset of R is 0 ≤ P (X ϵ E) ≤ 1 for each E ⊆ R o If E₁, E₂, E₃, … En are mutual exclusive then P (X ϵ (E₁ ⋃ E₂ ⋃ E₃ … ⋃ Ek) = P (X ϵ E₁) + P (X ϵ E₂) + P (X ϵ E₃) + … P(X ϵ Ek) Descriptive statistic Numerical summary of data • ➔given a sample of observations x₁, x₂, x₃, … xn with X as a random variable • SAMPLE MEAN ➔ 𝑥̅= ∑ 𝑥𝑖 𝑛 𝑖=1 𝑛 • SAMPLE VARIANCE ➔ 𝑠2= 𝛴𝑖=1 𝑛 (𝑥𝑖−𝑥̅)2 𝑛−1 • SAMPLE STANDARD DEVIATION ➔ 𝑠=√𝛴𝑖=1 𝑛 (𝑥𝑖−𝑥̅)2 𝑛−1 • MEDIAN (only for continuous probability functions) ➔ P(X ≤ m) = P(X ≥ m) = ½ • QUARTILES: correspond to 3 points (Q₁, median, Q₃) that divide the dataset in 4 equal groups (each group contains a quarter of the data) Other summaries of data • MOVING AVERAGE: is another method of batching data, but instead of considering separate batches that do not overlap I can consider the moving average of windows of size b → I have j batches of size b and for each of them I consider the moving average as 𝑥̅𝑗= ∑ 𝑥(𝑗−1)+𝑖 𝑏 𝑖=1 𝑏 o Example: I have 1000 observations that I want to batch in samples of 10 data each batch → with this method I get 999 batches of 10 data each o In the new dataset I will get 999 values that correspond to the sample means of the overlapping batches o…