Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 7 Random Variables 7.1: Discrete and Continuous Random Variables Random Variables • A random variable is a variable whose value is a numerical outcome of a random phenomenon. – the basic units of sampling distributions. – 2 types: discrete and continuous Discrete Random Variables • A discrete random variable X has a countable number of possible values. • The probability distribution of a discrete random variable X lists the values and their probabilities. Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 … pk Probability Distribution • The probability pi must satisfy two requirements. – Every probability pi is an number between 0 and 1 – The sum of the probabilities is 1: p1 + p2 + p3 +…+pk = 1 • Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event. Example: Maturation of male college students • In an article in the journal Developmental Psychology (March 1986), a probability distribution for the age X (in years) when male college students began to shave regularly is shown: X 11 12 13 14 15 P(X) 0.013 0 0.027 0.067 0.213 X 16 17 18 19 ≥20 P(X) 0.267 0.240 0.093 0.067 0.013 Example • Page 470 #7.2 Example • Page 470 #7.4 Continuous Random Variables • A continuous random variable X takes on all values in an interval of numbers. • The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up that event. • All continuous probability distributions assign probability 0 to every individual outcome. Example: Violence in Schools • Page 476 #7.9 Example: Drugs in schools • An opinion poll asks a SRS of 1500 American adults what they consider to be the most serious problem facing our schools. Suppose that if we could ask all adults this question, 30% would say “drugs”. What is the probability that the poll result differs from the truth about the population by more than two percentage points? N(.3, 0.0118) Chapter 7 Random Variables 7.2: Means and Variances of Random Variables Activity 7B • Page 481 Mean and expected Value • Mean of a probability distribution is denoted by µ, or µx. • The mean of the random variable, X is often referred to as the expected value of X. Mean of a Discrete Random Variable • Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 … pk To find the mean of X, multiply each possible value by its probability, then add all the products. Example • Page 486 #7.24 Example • Using the data from the “Maturation of male college students” example, find and interpret the mean. Variance of a Discrete Random Variable • Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 … pk and that µ is the mean of X. The variance of X is σx2 = Σ(x1 - µx)2pi. The standard deviation σx of X is the square root of the variance. Example • Page 486 #7.28 Technology Tip • To find µx and σx: Example • Using the data from the “Maturation of male college students” example, find the standard deviation. • Use the empirical rule to determine if the “Maturation of male college students” data is normally distributed. Sampling Distributions • The sampling distributions of statistics are just the probability distributions of these random variables. Law of Large Numbers • The average of a randomly selected sample from a large population is likely to be close to the average of the whole population. Law of Large Numbers • What is the mean of rolling 3 dice? Example: Emergency Evacuations • A panel of meteorological and civil engineers studying emergency evacuation plans for Florida’s Gulf Coast in the event of a hurricane has estimated that it take between 13 and 18 hours to evacuate people living in low-lying land, with the probabilities shown in the table. Let X = the time it takes a randomly selected person living in low-lying land in Florida to evacuate. Time to Evacuate Probability (nearest hour) 13 0.04 14 0.25 15 0.40 16 0.18 17 0.10 18 0.03 • Is X a discrete random variable or a continuous random variable? • Sketch a probability histogram for this data. • Find and interpret the mean. • Find the standard deviation. • Weather forecasters say that they cannot accurately predict a hurricane landfall more than 14 hours in advance. Find the probability that all residents of low-lying areas are evacuated safely if the Gulf Coast Civil Engineering Department waits until the 14-hour warning before beginning evacuation. Law of Small Numbers • Gambler’s Fallacy is the belief that every segment of a random sequence should reflect the true proportion. • This is a myth. There is no law of small numbers! Rules for Means • Rule 1: If X is a random variable and a and b are fixed numbers, then μa+bx = a + bμx • Rule 2: If X and Y are random variables, then μX + Y = μX + μY Rules for Variances • Rule 1: If X is a random variable and a and b are fixed numbers, then σ2a+bx = b2σ2x • Rule 2: If X and Y are independent random variables, then σ2X + Y = σ2x + σ2Y σ2X - Y = σ2x + σ2Y Means and Variances • Variances add, standard deviations don’t. • These rules can extend for more than 2 random variables…just follow the pattern. Combining Normal Random Variables • Any linear combination of independent Normal random variables is also Normally distributed. Example • Page 499 #7.38 • Page 499 #7.40 • Page 500 #7.42 Example • Page 501 #7.44 Example • Page 503 #7.50 Example • Page 503 #7.52