Experimental Design and Statistics - AGA47A

Transcription

1 Experimental Design and Statistics - AGA47A Czech University of Life Sciences in Prague Department of Genetics and Breeding Fall/Winter 2014/2015 Matúš Maciak (@ A 211) Office Hours: M 14:00 15:30 W 15:30 17:00 (or by appointment) 1 / 15

2 Brief Overview Classical vs. Geometrical Concept Classical Probability Concept Discrete Random Variables Geometrical Probability Concept Continuous Random Variables N 2 / 15

3 Brief Overview Discrete Random Variables Probability Mass Function: P[X = x] non-negative function with peaks (bars) located at outcome values; the size of each peak (bar) corresponds with the given probability; the sum of all peaks sizes (bars) equals always to one; Cumulative Probability Function: F (x) = P[X x] non-negative, non-decreasing and piece-wise constant function; the size of each jump is equal to the corresponding probability; it zero by definition on left, and equal to one on the right; Probability Cumulative Probability Random Variable Values Random Variable Values 3 / 15

4 Brief Overview Discrete Random Variables Probability Mass Function: P[X = x] non-negative function with peaks (bars) located at outcome values; the size of each peak (bar) corresponds with the given probability; the sum of all peaks sizes (bars) equals always to one; Cumulative Probability Function: F (x) = P[X x] non-negative, non-decreasing and piece-wise constant function; the size of each jump is equal to the corresponding probability; it zero by definition on left, and equal to one on the right; Probability Cumulative Probability Random Variable Values Random Variable Values Mutual relationship (c.p.f. definition) is given by: F (x) = P[X x] = P[X = x i] i; x i x 3 / 15

5 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 4 Bars 4 / 15

6 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 40 Bars 4 / 15

7 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 80 Bars 4 / 15

8 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 200 Bars 4 / 15

9 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 500 Bars 4 / 15

10 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 800 Bars 4 / 15

11 Introduction to Continuous Random Variables From Discrete to Continuous Frequency 2000 Bars 4 / 15

12 Introduction to Continuous Random Variables From Discrete to Continuous Density 4 / 15

13 Introduction to Continuous Random Variables Discrete vs. Continuous Variable Random variable X with some discrete distribution... 5 / 15

14 Introduction to Continuous Random Variables Discrete vs. Continuous Variable Random variable X with some discrete distribution... What is the Cumulative Probability Function F (x) = P[X x]? 5 / 15

15 Introduction to Continuous Random Variables Discrete vs. Continuous Variable Random variable X with some discrete distribution... What is the Cumulative Probability Function F (x) = P[X x]? Random variable X with some continuous distribution... 5 / 15

16 Introduction to Continuous Random Variables Discrete vs. Continuous Variable Random variable X with some discrete distribution... What is the Cumulative Probability Function F (x) = P[X x]? Random variable X with some continuous distribution... What is the Cumulative Probability Function F (x) = P[X x]? 5 / 15

17 Introduction to Continuous Random Variables Continuous Random Variable it is a random variable which can take any possible value from some given interval (or a set, or set of intervals respectively); 6 / 15

18 Introduction to Continuous Random Variables Continuous Random Variable it is a random variable which can take any possible value from some given interval (or a set, or set of intervals respectively); it is characterized by some non-negative density function f (x) 0, such that f (x)dx = 1; 6 / 15

19 Introduction to Continuous Random Variables Continuous Random Variable it is a random variable which can take any possible value from some given interval (or a set, or set of intervals respectively); it is characterized by some non-negative density function f (x) 0, such that f (x)dx = 1; the corresponding cumulative probability function is again defined by F (x) = P[X x]; It holds, that F = f and F (x) = x f (u)du; 6 / 15

20 Introduction to Continuous Random Variables Continuous Random Variable it is a random variable which can take any possible value from some given interval (or a set, or set of intervals respectively); it is characterized by some non-negative density function f (x) 0, such that f (x)dx = 1; the corresponding cumulative probability function is again defined by F (x) = P[X x]; It holds, that F = f and F (x) = x f (u)du; the same random variable characteristics are used again using the same definitions (mean, variance, etc.); 6 / 15

21 Introduction to Continuous Random Variables Continuous Random Variable it is a random variable which can take any possible value from some given interval (or a set, or set of intervals respectively); it is characterized by some non-negative density function f (x) 0, such that f (x)dx = 1; the corresponding cumulative probability function is again defined by F (x) = P[X x]; It holds, that F = f and F (x) = x f (u)du; the same random variable characteristics are used again using the same definitions (mean, variance, etc.); some similarities as well as obvious differences can be observed and between discrete and continuous random variables; 6 / 15

22 Statistics some data we collected as results of some unknown random mechanism; Introduction to Continuous Random Variables Random Variable Characteristics Probability random mechanism that we want to learn something about; We have to distinguish whether there is a discrete or continuous nature behind. Discrete Sample X 1,..., X n; Sample Mean: n X n = 1 X i n i=1 Sample Variance: n sn 2 = 1 (X i X n) 2 n 1 i=1 S = {x 1, x 2,...} Expectation: N E(X) = x i P[X = x i ] i=1 Variance: N Var(X) = P[X = x i ] (x i E(X)) 2 i=1 7 / 15

23 Statistics some data we collected as results of some unknown random mechanism; Introduction to Continuous Random Variables Random Variable Characteristics Probability random mechanism that we want to learn something about; We have to distinguish whether there is a discrete or continuous nature behind. Discrete Continuous Sample X 1,..., X n; Sample Mean: n X n = 1 X i n i=1 Sample Variance: n sn 2 = 1 (X i X n) 2 n 1 i=1 S = {x 1, x 2,...} Expectation: N E(X) = x i P[X = x i ] i=1 Variance: N Var(X) = P[X = x i ] (x i E(X)) 2 i=1 7 / 15

24 Statistics some data we collected as results of some unknown random mechanism; Introduction to Continuous Random Variables Random Variable Characteristics Probability random mechanism that we want to learn something about; We have to distinguish whether there is a discrete or continuous nature behind. Discrete Continuous Sample X 1,..., X n; Sample Mean: n X n = 1 X i n i=1 S = {x 1, x 2,...} Expectation: N E(X) = x i P[X = x i ] i=1 Population S; Sample Variance: n sn 2 = 1 (X i X n) 2 n 1 i=1 Variance: N Var(X) = P[X = x i ] (x i E(X)) 2 i=1 7 / 15

25 Statistics some data we collected as results of some unknown random mechanism; Introduction to Continuous Random Variables Random Variable Characteristics Probability random mechanism that we want to learn something about; We have to distinguish whether there is a discrete or continuous nature behind. Discrete Continuous Sample X 1,..., X n; Sample Mean: n X n = 1 X i n i=1 S = {x 1, x 2,...} Expectation: N E(X) = x i P[X = x i ] i=1 Population S; Expectation: E(X) = xf (x)dx Sample Variance: n sn 2 = 1 (X i X n) 2 n 1 i=1 Variance: N Var(X) = P[X = x i ] (x i E(X)) 2 i=1 7 / 15

26 Statistics some data we collected as results of some unknown random mechanism; Introduction to Continuous Random Variables Random Variable Characteristics Probability random mechanism that we want to learn something about; We have to distinguish whether there is a discrete or continuous nature behind. Discrete Continuous Sample X 1,..., X n; Sample Mean: n X n = 1 X i n i=1 Sample Variance: n sn 2 = 1 (X i X n) 2 n 1 i=1 S = {x 1, x 2,...} Expectation: N E(X) = x i P[X = x i ] i=1 Variance: Population S; Expectation: E(X) = xf (x)dx Variance: Var(X) = f (x)(x E(X)) 2 dx 7 / 15

27 Continuous Distributions in Statistics Various Continuous Distributions basically, any non-negative function f (x) 0, such that f (x)dx = 1, defines some continuous distribution; almost all of them are not important, however, there some crucial ones; 8 / 15

28 Continuous Distributions in Statistics Various Continuous Distributions basically, any non-negative function f (x) 0, such that f (x)dx = 1, defines some continuous distribution; almost all of them are not important, however, there some crucial ones; 8 / 15

29 Continuous Distributions in Statistics Various Continuous Distributions basically, any non-negative function f (x) 0, such that f (x)dx = 1, defines some continuous distribution; almost all of them are not important, however, there some crucial ones; 8 / 15

30 Continuous Distributions in Statistics Common Continuous Distributions Some most common continuous distributions: Uniform distribution; Laplace distribution; Gamma distribution; Weibull distribution; Exponential distribution; Beta distribution; χ 2 distribution; Student t distribution; Fisher distribution; Cauhy distribution; Gaussian (Normal) distribution; 9 / 15

31 Continuous Distributions in Statistics Common Continuous Distributions Some most common continuous distributions: Uniform distribution; Laplace distribution; Gamma distribution; Weibull distribution; Exponential distribution; Beta distribution; χ 2 distribution; Student t distribution; Fisher distribution; Cauhy distribution; Gaussian (Normal) distribution; 9 / 15

32 Continuous Distributions in Statistics Common Continuous Distributions Some most common continuous distributions: Uniform distribution; Laplace distribution; Gamma distribution; Weibull distribution; Exponential distribution; Beta distribution; χ 2 distribution; Student t distribution; Fisher distribution; Cauhy distribution; Gaussian (Normal) distribution; 9 / 15

33 Continuous Distributions in Statistics Common Continuous Distributions Some most common continuous distributions: Uniform distribution; Laplace distribution; Gamma distribution; Weibull distribution; Exponential distribution; Beta distribution; χ 2 distribution; Student t distribution; Fisher distribution; Cauhy distribution; Gaussian (Normal) distribution; 9 / 15

34 Continuous Distributions in Statistics Common Continuous Distributions Some most common continuous distributions: Uniform distribution; Laplace distribution; Gamma distribution; Weibull distribution; Exponential distribution; Beta distribution; χ 2 distribution; Student t distribution; Fisher distribution; Cauhy distribution; Gaussian (Normal) distribution; and many others... 9 / 15

35 Continuous Distributions in Statistics Uniform Continuous Distribution 10 / 15

36 Continuous Distributions in Statistics Uniform Continuous Distribution in some sense the simplest continuous distribution X Unif (a, b); it is defined on some interval (a, b) R; it also holds that f (x) = 1 a+b, E(X) = and Var(X) = (b a)2 ; b a / 15

37 Continuous Distributions in Statistics Exponential Distribution 11 / 15

38 Continuous Distributions in Statistics Exponential Distribution it is used to model times between some event occurrences: X Exp(λ); it is defined on positive values [0, ) R, for λ > 0; it holds that, f (x) = λe λx, E(X) = 1 λ and Var(X) = 1 λ 2 ; 11 / 15

39 Relationship Between Poisson Counts and Exponential Times Exponential Poisson Distribution How many occurrences did we observe? What is the distribution? 12 / 15

40 Relationship Between Poisson Counts and Exponential Times Exponential Poisson Distribution How many occurrences did we observe? What is the distribution? What are the times between occurrences? What is the distribution? 12 / 15

41 Relationship Between Poisson Counts and Exponential Times Exponential Poisson Distribution How many occurrences did we observe? What is the distribution? What are the times between occurrences? What is the distribution? 12 / 15

42 Gaussian Distribution The Bell Curve in Statistics 13 / 15

43 Gaussian Distribution Normal (Gaussian) Distribution 14 / 15

44 Gaussian Distribution Normal (Gaussian) Distribution it is used to model / 15

45 Gaussian Distribution Normal (Gaussian) Distribution it is used to model... anything & everything (X N(µ, σ 2 )); 14 / 15

46 Gaussian Distribution Normal (Gaussian) Distribution it is used to model... anything & everything (X N(µ, σ 2 )); it is defined (positive) for any x R; 14 / 15

47 Gaussian Distribution To be continued... briefly on statistical inference; inference on population mean; two sample mean; / 15