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Models of Discrete Random Variables Models of Continuous Random Variables Jiřı́ Neubauer Department of Econometrics FVL UO Brno office 69a, tel. 973 442029 email:[email protected] Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution Definition If the probability density function of X is 1 α < x < β, f (x) = β−α 0 otherwise, where α, β ∈ R, α < β, then X is said to have a uniform distribution on (α, β), written X ∼ R(α, β) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution The cumulative distribution function can be calculated as Zx F (x) = Zx f (t)dt = −∞ 1 x −α dt = · · · = β−α β−α for α < x < β. α Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution The cumulative distribution function can be calculated as Zx F (x) = Zx f (t)dt = −∞ 1 x −α dt = · · · = β−α β−α for α < x < β. α We obtain 0 x −α F (x) = β−α 1 Jiřı́ Neubauer x ≤ α, α < x < β, x ≥ β. Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution Figure: The probability density and the cumulative distribution function R(α, β) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution The table summarizes some basic information about the uniform distribution. E (X ) α+β 2 D(X ) 1 12 (β − α)2 α3 (X ) α4 (X ) quantiles xP Me(X ) 0 −1.2 α + P(β − α) α+β 2 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution The table summarizes some basic information about the uniform distribution. E (X ) α+β 2 D(X ) 1 12 (β − α)2 α3 (X ) α4 (X ) quantiles xP Me(X ) 0 −1.2 α + P(β − α) α+β 2 Examples: a time we wait for a bus (buses go regularly every 10 minutes), a time we wait for a supply of bread in a grocery store (supplies are regular), calculation rounding mistakes, . . . Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution Using: P(X ≤ x0 ) = F (x0 ) = x0 −α β−α for x0 ∈ (α, β) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Uniform Distribution Using: P(X ≤ x0 ) = F (x0 ) = x0 −α β−α for x0 ∈ (α, β) P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = Jiřı́ Neubauer x2 −α β−α − x1 −α β−α for x1 , x2 ∈ (α, β) Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Trams go regularly every 10 minutes. The passenger comes to the tram-stop at the arbitrary time. The random variable X is the time he/she has to wait for a tram. Find the probability density function and the distribution function of X . Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Trams go regularly every 10 minutes. The passenger comes to the tram-stop at the arbitrary time. The random variable X is the time he/she has to wait for a tram. Find the probability density function and the distribution function of X . What is the probability that the passenger will wait at most 3 minutes, at least 5 minutes, exactly 7 minutes. Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Trams go regularly every 10 minutes. The passenger comes to the tram-stop at the arbitrary time. The random variable X is the time he/she has to wait for a tram. Find the probability density function and the distribution function of X . What is the probability that the passenger will wait at most 3 minutes, at least 5 minutes, exactly 7 minutes. Calculate the mean, the median, the variance, the standard deviation and the 90 % quantile. Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The random variable we can describe by the uniform distribution X ∼ R(0, 10). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The random variable we can describe by the uniform distribution X ∼ R(0, 10). The probability density function is 1 0 < x < 10, 10 f (x) = 0 otherwise, Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The random variable we can describe by the uniform distribution X ∼ R(0, 10). The probability density function is 1 0 < x < 10, 10 f (x) = 0 otherwise, the cumulative distribution function 0 x F (x) = 10 1 Jiřı́ Neubauer is x ≤ 0, 0 < x < 10, x ≥ 10. Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Figure: The probability density and the cumulative distribution function R(0, 10) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The probability that the passenger will wait at most 3 minutes R3 1 P(X ≤ 3) = 10 dx = 0 1 3 10 [x]0 = 0.3 using the distribution function 3 P(X ≤ 3) = F (3) = 10 = 0.3 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The probability that the passenger will wait at most 3 minutes R3 1 P(X ≤ 3) = 10 dx = 1 3 10 [x]0 0 = 0.3 using the distribution function 3 P(X ≤ 3) = F (3) = 10 = 0.3 at least 5 minutes R10 1 P(X ≥ 5) = 10 dx = 5 1 10 10 [x]5 = 0.5 P(X ≥ 5) = 1 − P(X < 5) = 1 − P(X ≤ 5) = 1 − F (5) = 5 = 1 − 10 = 0.5 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The probability that the passenger will wait at most 3 minutes R3 1 P(X ≤ 3) = 10 dx = 1 3 10 [x]0 0 = 0.3 using the distribution function 3 P(X ≤ 3) = F (3) = 10 = 0.3 at least 5 minutes R10 1 P(X ≥ 5) = 10 dx = 5 1 10 10 [x]5 = 0.5 P(X ≥ 5) = 1 − P(X < 5) = 1 − P(X ≤ 5) = 1 − F (5) = 5 = 1 − 10 = 0.5 exactly 7 minutes P(X = 7) = 0 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example the mean E (X ) = α+β 2 = 10 2 =5 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example α+β 10 2 = 2 =5 10 Me(X ) = α+β 2 = 2 = the mean E (X ) = the median Jiřı́ Neubauer 5 Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example α+β 10 2 = 2 =5 10 median Me(X ) = α+β 2 = 2 =5 1 1 variance D(X ) = 12 (β − α)2 = 12 (10 the mean E (X ) = the the Jiřı́ Neubauer − 0)2 = 100 12 = 8.333 Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example α+β 10 2 = 2 =5 10 median Me(X ) = α+β 2 = 2 =5 1 1 variance D(X ) = 12 (β − α)2 = 12 (10 the mean E (X ) = the − 0)2 = 100 12 = 8.333 q p the standard variation σ = D(X ) = 100 12 = 2.887 the Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example α+β 10 2 = 2 =5 10 median Me(X ) = α+β 2 = 2 =5 1 1 variance D(X ) = 12 (β − α)2 = 12 (10 the mean E (X ) = the − 0)2 = 100 12 = 8.333 q p the standard variation σ = D(X ) = 100 12 = 2.887 the 90 % quantile x0.90 = α + 0.90(β − α) = 0.9 · 10 = 9 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution Definition If the probability density function of X is ( 1 − x−α e δ f (x) = δ 0 x > α, x ≤ α, where α ∈ R, δ > 0, then X is said to have an exponential distribution with parameters α and δ, written X ∼ Ex(α, δ). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution The cumulative distribution function is x−α 1 − e− δ F (x) = 0 Jiřı́ Neubauer x > α, x ≤ α. Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution Figure: The probability density and the cumulative distribution function Ex(α, δ) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution The table summarizes some basic information about the exponential distribution. E (X ) D(X ) α3 (X ) α4 (X ) quantiles xP Me(X ) α+δ δ2 2 6 α − δ ln(1 − P) α + δ ln 2 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution The table summarizes some basic information about the exponential distribution. E (X ) D(X ) α3 (X ) α4 (X ) quantiles xP Me(X ) α+δ δ2 2 6 α − δ ln(1 − P) α + δ ln 2 Examples: the queuing theory, the reliability theory, the renewal theory, a time we wait for service, a product lifetime, . . . Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution Using: P(X ≤ x0 ) = F (x0 ) = 1 − e − Jiřı́ Neubauer x0 −α δ for x0 > α Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Exponential distribution Using: P(X ≤ x0 ) = F (x0 ) = 1 − e − x0 −α δ for x0 > α P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = e − Jiřı́ Neubauer x1 −α δ − e− x2 −α δ for x1 , x2 > α Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example It has been found out the time we have to wait for a waiter is a random variable which has a exponential distribution with the mean 5 minutes and the standard deviation 2 minutes. Plot the probability density function and the distribution function. What is the probability that we will wait at most 5 minutes? Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The mean a the variance of the exponential distribution are E (X ) = α + δ and D(X ) = δ 2 , thus α+δ=5 ⇒ α = 3, δ = 2 δ=2 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The mean a the variance of the exponential distribution are E (X ) = α + δ and D(X ) = δ 2 , thus α+δ=5 ⇒ α = 3, δ = 2 δ=2 X ∼ Ex(3, 2) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Figure: The probability density and the cumulative distribution function Ex(α, δ) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The probability that we will wait at most 5 minutes is P(X ≤ 5) = F (5) = 1 − e − Jiřı́ Neubauer 5−3 2 = 1 − e −1 = 0.632. Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Definition If X has the probability density function f (x) = (x−µ)2 1 √ e − 2σ2 σ 2π for x ∈R where µ ∈ R, σ 2 > 0, it is said to have a normal distribution with parameters µ and σ 2 , written X ∼ N(µ, σ 2 ). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The cumulative distribution function is Zx F (x) = f (t)dt = −∞ 1 √ σ 2π Jiřı́ Neubauer Zx e− (t−µ)2 2σ 2 dt for x ∈R −∞ Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Figure: The probability density and the cumulative distribution function N(µ, σ 2 ) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The table summarizes some basic information about the normal distribution. E (X ) D(X ) α3 (X ) α4 (X ) quantiles xP Me(X ) Mo(X ) µ σ2 0 0 µ + σuP 1 µ µ 1 quantile of the standard normal distribution N(0, 1) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The normally distributed random variable fulfils : P(µ − σ < X < µ + σ) = 0.683 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The normally distributed random variable fulfils : P(µ − σ < X < µ + σ) = 0.683 P(µ − 2σ < X < µ + 2σ) = 0.954 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The normally distributed random variable fulfils : P(µ − σ < X < µ + σ) = 0.683 P(µ − 2σ < X < µ + 2σ) = 0.954 P(µ − 3σ < X < µ + 3σ) = 0.997 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Let us have the random variable X ∼ N(µ, σ 2 ). The transformed random variable U X −µ U= σ has the normal distribution with the mean 0 and the variance 1 (the standard normal distribution U ∼ N(0, 1)). Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The probability density function is u2 1 φ(u) = √ e − 2 2π for u ∈ R, the cumulative distribution function is Zu Φ(u) = −∞ 1 φ(t)dt = √ 2π Jiřı́ Neubauer Zu t2 e − 2 dt for u∈R −∞ Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Figure: The probability density and the cumulative distribution function N(0, 1) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The table summarizes some basic information about the standard normal distribution. E (X ) D(X ) α3 (X ) α4 (X ) kvantily xP Me(X ) Mo(X ) 0 1 0 0 uP 1 0 0 1 the values are tabulated, for P < 0.5 is uP = −u1−P Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The values of the cumulative distribution function for positive values are tabulated, for negative values we can write Φ(−u) = 1 − Φ(u). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution The values of the cumulative distribution function for positive values are tabulated, for negative values we can write Φ(−u) = 1 − Φ(u). If X ∼ N(µ, σ 2 ), U ∼ N(0, 1), then the cumulative distribution function of the random variable X we can obtain using cumulative distribution function of U. x0 −µ F (x0 ) = P(X ≤ x0 ) = P(X − µ ≤ x0 − µ) = P X −µ ≤ = σ σ x0 −µ x0 −µ =P U ≤ σ =Φ σ Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Quantiles of X (quantiles of U are tabulated): xP − µ = Φ(uP ), F (xP ) = Φ σ thus uP = xP − µ ⇒ xP = µ + σuP . σ Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Using: P(X ≤ x0 ) = F (x0 ) = Φ x0 −µ σ Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Normal Distribution Using: P(X ≤ x0 ) = F (x0 ) = Φ x0 −µ σ P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = Φ Jiřı́ Neubauer x2 −µ σ −Φ x1 −µ σ Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example During quality control we say that the component is acceptable if it’s size is within the limits 26–27 mm. The size of the component has a normal distribution with the mean µ = 26.4 mm and the standard deviation σ = 0.2 mm. What is the probability that the size of the component is within the given limits? Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The random variable X ∼ N(26.4; 0.22 ). − Φ 26−26.4 = P(26 ≤ X ≤ 27) = F (27) − F (26) = Φ 27−26.4 0.2 0.2 = Φ(3) − Φ(−2) = Φ(3) − (1 − Φ(2)) = = 0.99865 − (1 − 0.97725) = 0.9759 Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Figure: The probability density and the cumulative distribution function N(26.4; 0.04) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution Let us assume that X is a non-negative random variable. If a random variable ln X has a normal distribution N(µ, σ 2 ), then X has a log-normal distribution LN(µ, σ 2 ). Definition If the probability density function of X (ln x−µ)2 1 √ e − 2σ2 f (x) = xσ 2π 0 x > 0, x ≤ 0, where µ ≥ 0, σ > 0, then X is said to have a log-normal distribution with parameters µ and σ 2 , written X ∼ LN(µ, σ 2 ). Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution The table summarizes some basic information about the log-normal distribution. E (X ) D(X ) α3 (X ) α4 (X ) quantiles xP 2 e µ+σ /2 e 2µ ω(ω−1) √ ω−1(ω+2) ω 4 +2ω 3 +3ω 2 −6 e µ+σuP where ω = e σ 2 Jiřı́ Neubauer Models of Continuous Random Variables Mo(X ) e µ−σ 2 The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution The table summarizes some basic information about the log-normal distribution. E (X ) D(X ) α3 (X ) α4 (X ) quantiles xP 2 e µ+σ /2 e 2µ ω(ω−1) √ ω−1(ω+2) ω 4 +2ω 3 +3ω 2 −6 e µ+σuP 2 Mo(X ) e µ−σ 2 where ω = e σ Examples: model of entry and wages distributions, a time of renewals, repairs, the theory of non-coherent particles, . . . Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution If the random variable X has the log-normal distribution X ∼ LN(µ, σ), then the transformed random variable U= ln X − µ σ has the standard normal distribution U ∼ N(0, 1). Jiřı́ Neubauer Models of Continuous Random Variables The The The The The Models of Discrete Random Variables Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution If the random variable X has the log-normal distribution X ∼ LN(µ, σ), then the transformed random variable U= ln X − µ σ has the standard normal distribution U ∼ N(0, 1). We can write ln x0 − µ F (x0 ) = Φ = Φ(u), σ where Φ(u) is the cumulative distribution function N(0, 1). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution Using: P(X ≤ x0 ) = F (x0 ) = Φ ln x0 −µ σ Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Log-normal Distribution Using: P(X ≤ x0 ) = F (x0 ) = Φ ln x0 −µ σ P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = Φ Jiřı́ Neubauer ln x2 −µ σ −Φ ln x1 −µ σ Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example We suppose that distance between vehicles on the highway (in seconds) is a random variable which is possible to describe by the log-normal distribution with parameters µ = 1.27 a σ 2 = 0.49. What is the probability that the distance will be from 4 till 5 seconds? Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example Figure: The probability density and the cumulative distribution function LN(1.27; 0.7) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution Example The probability is P(4 ≤ X ≤ 5) = F (5) − F (4) = Φ ln 5−1.27 −Φ 0.7 = 0.68613 − 0.56597 = 0.12016 Jiřı́ Neubauer ln 4−1.27 0.7 Models of Continuous Random Variables = Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Pearson χ2 Distribution Definition Let us assume that U1 , U2 , . . . , Uν are independent normally distributed random variables (N(0, 1)). The random variable χ2 = U12 + U22 + · · · + Uν2 , has a χ2 -distribution with ν degrees of freedom. The parameter ν (the number of freedom) usually represents the number of independent observation reduced by the number of linear conditions. Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Pearson χ2 Distribution Figure: The probability density and the cumulative distribution function χ2 (5) and χ2 (16) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Pearson χ2 Distribution In the future the quantiles of the χ2 distribution will be useful. They are usually tabulated for various values P and degrees of freedom ν ≤ 30. For ν > 30 is possible to use an approximation χ2P (ν) ≈ 2 1 √ 2ν − 1 + uP , 2 where uP is the quantile of N(0, 1). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Student distribution t(ν) Definition If a random variable U has a standard normal distribution U ∼ N(0, 1), a random variable χ2 has a Pearson distribution χ2 ∼ χ2 (ν) and if U and χ2 are independent, then a random variable U t=q χ2 ν has a Student distribution with ν degrees of freedom, written t ∼ t(ν). Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Student distribution t(ν) Figure: The probability density and the cumulative distribution function t(2) and t(20) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Student distribution t(ν) The probability density function is symmetric with the mean E (t) = 0. Quantiles of the Student distribution are tabulated for ν ≤ 30 and P > 0,5, for P < 0, 5 is tP = −t1−P . Whether ν > 30, we can use an approximation tp ≈ uP . Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Fisher-Snedecor Distribution F (ν1 , ν2 ) Definition If a random variable χ21 has χ21 ∼ χ2 (ν1 ) with ν1 degrees of freedom and a random variable χ22 has χ22 ∼ χ2 (ν2 ) with ν2 degrees of freedom and they are independent, then a random variable F = χ21 χ22 : ν1 ν2 has a Fisher-Snedecor distribution with ν1 and ν2 degrees of freedom, written F ∼ F (ν1 , ν2 ) . Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Fisher-Snedecor Distribution F (ν1 , ν2 ) Figure: The probability density and the cumulative distribution function F (30, 20) and F (3, 50) Jiřı́ Neubauer Models of Continuous Random Variables Models of Discrete Random Variables The The The The The Uniform Distribution Exponential distribution Normal Distribution Log-normal Distribution Pearson, the Student and the Fisher-Snedecor Distribution The Fisher-Snedecor Distribution F (ν1 , ν2 ) The Fischer-Snedecor distribution is asymmetric. Quantiles of F distribution are tabulated for P > 0,5, for P < 0,5 we can use the formula 1 FP (ν1 , ν2 ) = . F1−P (ν2 , ν1 ) Jiřı́ Neubauer Models of Continuous Random Variables