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Lecture 8: Special Probability Densities The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Lecture 8: Special Probability Densities Assist. Prof. Dr. Emel YAVUZ DUMAN Int. to Prob. Theo. and Stat & Int. to Probability Istanbul˙ K¨ult¨ur University Faculty of Engineering The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Outline 1 The Uniform Distribution 2 The Normal Distribution 3 The Normal Approximation to the Binomial Distribution 4 The Normal Approximation to the Poisson Distribution The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Outline 1 The Uniform Distribution 2 The Normal Distribution 3 The Normal Approximation to the Binomial Distribution 4 The Normal Approximation to the Poisson Distribution The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Definition 1 A random variable has a uniform distribution and it is refereed to as a continuous uniform random variable if and only if its probability density is given by 1 β α for α< x < β, u(x; α,β)= − (0 elsewhere. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Definition 1 A random variable has a uniform distribution and it is refereed to as a continuous uniform random variable if and only if its probability density is given by 1 β α for α< x < β, u(x; α,β)= − (0 elsewhere. The parameters α and β of this probability density are real constants, with α<β, The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Definition 1 A random variable has a uniform distribution and it is refereed to as a continuous uniform random variable if and only if its probability density is given by 1 β α for α< x < β, u(x; α,β)= − (0 elsewhere. The parameters α and β of this probability density are real constants, with α<β, and may be pictured as in the figure. u(x; α,β) 1 β α − x α β The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Theorem 2 Them mean and the variance of the uniform distribution are given by α + β 1 µ = and σ2 = (β α)2 2 12 − The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Theorem 2 Them mean and the variance of the uniform distribution are given by α + β 1 µ = and σ2 = (β α)2 2 12 − Proof. The mean: β x µ = E(X )= ∞ xu(x; α,β)dx = dx α β α Z−∞ Z − β x2 β2 α2 (β α)(β + α) = = − = − 2(β α) 2(β α) 2(β α) − α − − β + α = 2 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation The variance: β x2 E(X 2)= ∞ x2u(x; α,β)dx = dx α β α Z−∞ Z − β x3 β3 α3 (β α)(β2 + βα + α2) = = − = − 3(β α) 3(β α) 3(β α) − α − − 2 2 β + βα + α = 3 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation The variance: β x2 E(X 2)= ∞ x2u(x; α,β)dx = dx α β α Z−∞ Z − β x3 β3 α3 (β α)(β2 + βα + α2) = = − = − 3(β α) 3(β α) 3(β α) − α − − 2 2 β + βα + α = 3 β2 + βα + α2 (β + α)2 σ2 = E(X 2) (E(X ))2 = − 3 − 22 4β2 + 4βα + 4α2 3β2 6αβ 3α2 = − − − 12 β2 2βα + α2 (β α)2 = − = − . 12 12 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Example 3 Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 am and 11 pm during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes. (a) Draw a graph of this distribution. (b) Show that the area of this uniform distribution is 1.00. (c) How long will a student typically have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times? (d) What is the probability a student will wait more than 25 minutes? (e) What is the probability a student will wait between 10 and 20 minutes? The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Solution. (a) The graph of this distribution: Probability 1 ¯ 30 0 = 0.03 − 0 10 20 30 Length of wait (minutes) The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation (b) The times students must wait for the bus is uniform over the interval from 0 minutes to 30 minutes, so in this case α = 0 and β = 30. 1 Area = (height)(base) (30 0) = 0 (30 0) − − or 30 1 Area = dx 30 0 Z0 − x 30 30 0 = = − = 1. 30 0 30 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation (c) The mean waiting time: α + β 0 + 30 µ = = = 15 2 2 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation (c) The mean waiting time: α + β 0 + 30 µ = = = 15 2 2 The standard deviation of the waiting time: (β α)2 (30 0)2 σ = − = − = 5√3 = 8.6603 r 12 r 12 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation (c) The mean waiting time: α + β 0 + 30 µ = = = 15 2 2 The standard deviation of the waiting time: (β α)2 (30 0)2 σ = − = − = 5√3 = 8.6603 r 12 r 12 P(X ) 0.03¯ 0 10 20 30 µ = 15 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation 30 1 x 30 5 1 ¯ (d) P(25 < X < 30) = 25 30 0 dx = 30 25 = 30 = 6 = 0.16 − R The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation 30 1 x 30 5 1 ¯ (d) P(25 < X < 30) = 25 30 0 dx = 30 25 = 30 = 6 = 0.16 − R P(X ) Area= 0.16¯ 0.03¯ 0 10 2025 30 µ = 15 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation 20 1 x 20 10 1 ¯ (e) P(10 X 20) = 10 30 0 dx = 30 10 = 30 = 3 = 0.3 ≤ ≤ − R The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation 20 1 x 20 10 1 ¯ (e) P(10 X 20) = 10 30 0 dx = 30 10 = 30 = 3 = 0.3 ≤ ≤ − R P(X ) Area= 0.3¯ 0.03¯ 0 10 20 30 µ = 15 The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Outline 1 The Uniform Distribution 2 The Normal Distribution 3 The Normal Approximation to the Binomial Distribution 4 The Normal Approximation to the Poisson Distribution The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation The normal distribution, which we shall study in this section, is in many ways the cornerstone of modern statistical theory. Definition 4 A random variable X has a normal distribution and it is referred to as a normal random variable if and only if its probability density is given by 1 1 ( x−µ )2 n(x; µ,σ)= e− 2 σ for < x < σ√2π − ∞ ∞ where σ > 0. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Characteristics of a Normal Distribution It is bell-shaped and has a single peak at the center of the distribution. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Characteristics of a Normal Distribution It is bell-shaped and has a single peak at the center of the distribution. It is symmetrical about the mean. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Characteristics of a Normal Distribution It is bell-shaped and has a single peak at the center of the distribution. It is symmetrical about the mean. It is asymptotic: The curve gets closer and closer to the x-axis but never actually touches it. To put it another way, the tails of the curve extend indefinitely in both directions. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Characteristics of a Normal Distribution It is bell-shaped and has a single peak at the center of the distribution. It is symmetrical about the mean. It is asymptotic: The curve gets closer and closer to the x-axis but never actually touches it. To put it another way, the tails of the curve extend indefinitely in both directions. The location of a normal distribution is determined by the mean, µ, the dispersion or spread of the distribution is determined by the standard deviation, σ. The Uniform Distribution The Normal Distribution The Normal Approximation to the Binomial Distribution The Normal Approximation Characteristics of a Normal Distribution It is bell-shaped and has a single peak at the center of the distribution.
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