Chapter 5 Continuous Random Variables Where Weve Been
Chapter 5: Continuous Random Variables
Where We’ve Been n n Using probability rules to find the probability of discrete events Examined probability models for discrete random variables Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 2
Where We’re Going n n n Develop the notion of a probability distribution for a continuous random variable Examine several important continuous random variables and their probability models Introduce the normal probability distribution Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 3
5. 1: Continuous Probability Distributions n n A continuous random variable can assume any numerical value within some interval or intervals. The graph of the probability distribution is a smooth curve called a ¡ ¡ ¡ probability density function, frequency function or probability distribution. Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 4
5. 1: Continuous Probability Distributions n There an infinite number of possible outcomes ¡ ¡ p(x) = 0 Instead, find p(a<x<b) Table Software Integral calculus) Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 5
5. 2: The Uniform Distribution n n X can take on any value between c and d with equal probability = 1/(d - c) For two values a and b Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 6
5. 2: The Uniform Distribution Mean: Standard Deviation: Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 7
5. 2: The Uniform Distribution Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10 x 18)? Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 8
5. 2: The Uniform Distribution Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10 x 18)? Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 9
5. 3: The Normal Distribution n Closely approximates many situations ¡ n Perfectly symmetrical around its mean The probability density function f(x): µ = the mean of x = the standard deviation of x = 3. 1416… e = 2. 71828 … Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 10
5. 3: The Normal Distribution n n Each combination of µ and produces a unique normal curve The standard normal curve is used in practice, based on the standard normal random variable z (µ = 0, = 1), with the probability distribution The probabilities for z are given in Table IV Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 11
5. 3: The Normal Distribution Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 12
5. 3: The Normal Distribution For a normally distributed random variable x, if we know µ and , So any normally distributed variable can be analyzed with this single distribution Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 13
5. 3: The Normal Distribution n n Say a toy car goes an average of 3, 000 yards between recharges, with a standard deviation of 50 yards (i. e. , µ = 3, 000 and = 50) What is the probability that the car will go more than 3, 100 yards without recharging? Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 14
5. 3: The Normal Distribution n n Say a toy car goes an average of 3, 000 yards between recharges, with a standard deviation of 50 yards (i. e. , µ = 3, 000 and = 50) What is the probability that the car will go more than 3, 100 yards without recharging? Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 15
5. 3: The Normal Distribution n To find the probability for a normal random variable … ¡ ¡ ¡ Sketch the normal distribution Indicate x’s mean Convert the x variables into z values Put both sets of values on the sketch, z below x Use Table IV to find the desired probabilities Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 16
5. 4: Descriptive Methods for Assessing Normality n If the data are normal ¡ ¡ A histogram or stem-and-leaf display will look like the normal curve The mean ± s, 2 s and 3 s will approximate the empirical rule percentages The ratio of the interquartile range to the standard deviation will be about 1. 3 A normal probability plot , a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 17
5. 4: Descriptive Methods for Assessing Normality Errors per MLB team in 2003 Mean: 106 Standard Deviation: 17 ¡ IQR: 22 ¡ ¡ 22 out of 30: 73% Frequency Histogram 10 9 8 7 6 5 4 3 2 1 0 28 out of 30: 93% Frequency 77 89. 8 102. 6 115. 4 128. 2 More Errors per team, 2003 30 out of 30: 100% Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 18
5. 4: Descriptive Methods for Assessing Normality 3 Normal Quantile 2 1 0 -1 A normal probability plot is a scatterplot with the ranked data on one axis and the expected zscores from a standard normal distribution on the other axis -2 -3 60 80 100 120 140 160 Errors Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 19
5. 5: Approximating a Binomial Distribution with the Normal Distribution n n Discrete calculations may become very cumbersome The normal distribution may be used to approximate discrete distributions ¡ n The larger n is, and the closer p is to. 5, the better the approximation Since we need a range, not a value, the correction for continuity must be used ¡ A number r becomes r+. 5 Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 20
5. 5: Approximating a Binomial Distribution with the Normal Distribution Calculate the mean plus/minus 3 standard deviations If this interval is in the range 0 to n, the approximation will be reasonably close Express the binomial probability as a range of values Find the z-values for each binomial value Use the standard normal distribution to find the probability for the range of values you calculated Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 21
5. 5: Approximating a Binomial Distribution with the Normal Distribution Flip a coin 100 times and compare the binomial and normal results Binomial: Normal: Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 22
5. 5: Approximating a Binomial Distribution with the Normal Distribution Flip a weighted coin [P(H)=. 4] 10 times and compare the results Binomial: Normal: Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 23
5. 5: Approximating a Binomial Distribution with the Normal Distribution Flip a weighted coin [P(H)=. 4] 10 times and compare the results Binomial: Normal: The more p differs from. 5, and the smaller n is, the less precise the approximation will be Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 24
5. 6: The Exponential Distribution n Probability Distribution for an Exponential Random Variable x ¡ Probability Density Function ¡ Mean: µ = ¡ Standard Deviation: = Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 25
5. 6: The Exponential Distribution Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60 Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 26
5. 6: The Exponential Distribution Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60 Mc. Clave: Statistics, 11 th ed. Chapter 5: Continuous Random Variables 27
- Slides: 27
