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Normal distribution – revision MAS113 Introduction to Probability and Statistics Dr Jonathan Jordan School of Mathematics and Statistics, University of Sheffield Spring Semester, 2017 Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Definition Definition If a random variable X has a normal distribution with mean µ and variance σ 2 , then its probability density function is given by 1 1 2 fX (x) = √ exp − 2 (x − µ) 2σ 2πσ 2 Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Definition Definition If a random variable X has a normal distribution with mean µ and variance σ 2 , then its probability density function is given by 1 1 2 fX (x) = √ exp − 2 (x − µ) 2σ 2πσ 2 We write X ∼ N(µ, σ 2 ), to mean “X has a normal distribution with mean µ and variance σ 2 .” Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Properties E (X ) = µ and Var(X ) = σ 2 Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Properties E (X ) = µ and Var(X ) = σ 2 If Z ∼ N(0, 1) Z has a standard normal distribution. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Properties E (X ) = µ and Var(X ) = σ 2 If Z ∼ N(0, 1) Z has a standard normal distribution. Distribution function of standard normal Z z t2 1 √ e − 2 dt Φ(z) = P(Z ≤ z) = 2π −∞ Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Properties E (X ) = µ and Var(X ) = σ 2 If Z ∼ N(0, 1) Z has a standard normal distribution. Distribution function of standard normal Z z t2 1 √ e − 2 dt Φ(z) = P(Z ≤ z) = 2π −∞ given by pnorm in R. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Standardisation If Z ∼ N(0, 1) then µ + σZ ∼ N(µ, σ 2 ). Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Standardisation If Z ∼ N(0, 1) then µ + σZ ∼ N(µ, σ 2 ). If X ∼ N(µ, σ 2 ) then X −µ ∼ N(0, 1). σ Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision The two-σ rule For a standard normal random variable Z , P(−1.96 ≤ Z ≤ 1.96) = 0.95. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision The two-σ rule For a standard normal random variable Z , P(−1.96 ≤ Z ≤ 1.96) = 0.95. Since E (Z ) = 0 and Var(Z ) = 1, the probability of Z being within two standard deviations of its mean value is approximately 0.95 (ie P(−2 ≤ Z ≤ 2) = 0.9545 to 4 d.p.). Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision The two-σ rule: general case If we now consider any normal random variable X ∼ N(µ, σ 2 ), the probability that it will lie within a distance of two standard deviations from its mean is approximately 0.95. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision The two-σ rule: general case If we now consider any normal random variable X ∼ N(µ, σ 2 ), the probability that it will lie within a distance of two standard deviations from its mean is approximately 0.95. This is straightforward to verify: Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision The two-σ rule: general case If we now consider any normal random variable X ∼ N(µ, σ 2 ), the probability that it will lie within a distance of two standard deviations from its mean is approximately 0.95. This is straightforward to verify: (on board) Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Convention In Statistics, there is a convention of using 0.05 as a threshold for a ‘small’ probability, though the choice of 0.05 is arbitrary. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics Normal distribution – revision Convention In Statistics, there is a convention of using 0.05 as a threshold for a ‘small’ probability, though the choice of 0.05 is arbitrary. However, the two-σ rule is an easy to remember fact about normal random variables, and can be a useful yardstick in various situations. Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics