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AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Sample spaces do not need to consist of number. Sample spaces consist of outcomes. If two fair coins are tossed, the sample space, or outcomes are π»π», π»π, ππ», ππ . In the study of probability, we are more interested the number of outcomes, or counts, in the sample space than we are the actual outcomes. In the study of discrete, or countable, random variables, we use the variable π₯ or π to describe the number of successful outcomes in an event. For instance, if a fair coin is tossed four times and the variable of interest is the number of times the coin lands on heads, then π₯ = 0, 1, 2, 3, ππ 4. If the coin is tossed four more times, chances are, the number of heads will be different, although π₯ = 0, 1, 2, 3, ππ 4. in this sense, we can think of π as a random variable since its values vary from trial to trial. AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Random Variable: A variable whose value is a numerical outcome of a random phenomenon As we progress from general rules of probability toward statistical inference, we will focus more on random variables. When a random variable, π₯ describes a random phenomenon, the sample space, π just lists the possible values of the random variable. In a probability model, each possible outcome must also have a corresponding probability. Consider the example of tossing four coins & the variable of interest being the # of times the coins lands on heads. Value of π₯ 0 1 2 3 4 π π₯ AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Discrete Random Variables: A discrete random variable describes a phenomenon that is countable. In a probability distribution, the sum of all the probabilities must add to 1 and each individual probability must lie between 0 and 1 inclusive, or 0 β€ π πΈ β€ 1. This can be summarized as Value of π₯ π₯1 π₯2 π₯3 β¦ π₯π π πΈ π1 π2 π3 β¦ ππ AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables This data can also be summarized by using a probability histogram. Probability 0 1 2 3 4 AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Continuous Random Variables: A continuous random variable, π₯ takes all values in an interval of numbers. The probability distribution of π₯ is described by a density curve. The probability of any event is the area under a density curve. A continuous random variable does not describe a countable phenomenon. β΄ it is not possible to summarize the probability distribution by using a chart as we can with a discrete random variable, however, we can with a histogram since a histogram evolves into a density curve. AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Ex. Randomly select an integer between 0 and 10 inclusive. π₯ describes a discrete random variable Ex. Randomly select a number between 0 and 10 inclusive. π₯ describes a continuous random variable In a continuous random variable distribution, it is impossible for π₯ to represent a single value. π₯ must describe a range of values. In fact, all continuous probability distributions assign probability 0 to every individual outcome. AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is the difference between Discrete & Continuous Random Variables 7.1 Discrete and Continuous Random Variables Normal Distributions as Probability Distributions π₯βπ π§= π Suppose 500 Lee County residents are asked what they consider to be the most significant problem facing Lee County Schools. Suppose if we had the resources and time to ask all registered voters in Lee County the same question & 42% would say the one-to-one laptop initiative. This 0.42 is known as a population parameter, and the proportion of the 500 that also say the one-to-one laptop initiative, π, is called a sample statistic. Suppose π is a random variable that is normally distributed and is described as π 0.42, 0.012 . Find each of the followingβ¦ a. b. π π < 0.40 The probability that the poll results differs from the population parameter by more than 2 percentage points AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables Homework: AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables Probability is the language used to describe long-run regular behavior of random phenomenon. The probability distribution of a random variable is an idealized relative frequency distribution. The mean, or expected value, of a random variable is denoted as ππ₯ . It describes the average of all possible values of π₯, but keep in mind that not all values of π₯ are equally likely. ππ₯ = π₯π ππ AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables AP Statistics Part III β Probability: Foundations for Inference Inhabitants 2 3 4 5 6 7 8 9 10 Proportion of homes AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables Statistical Estimation & the Law of Large Numbers Draw independent observations at random from any population with finite mean, π. Decide how accurately you would like to estimate π. As the number of observations drawn increases, the mean, π₯ of the observed values eventually approaches the mean π of the population as closely as you specified and then stays that close. What does this mean? If you want your sample mean to be a true measure of the population mean, continue to draw larger samples from the same population and the sample mean will approach the true population mean. AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables The Law of Large Numbers governs many things such asβ¦ β’ β’ β’ β’ Casinos Insurance companies Restaurants Budgeting AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables Rules for Means β’ Rule 1: If π₯ is a random variable and π & π are fixed numbers, then ππ+ππ₯ = π + πππ₯ β’ Rule 2: If π₯ & π¦ are random variables, then ππ₯+π¦ = ππ₯ + ππ¦ AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables Both rules can be illustrated through simulation. Rule 1: Suppose each employee at a company earns 3% increase in pay for the coming year. Also, each person will receive a $1500 bonus at the end of the year. Rule 2: Suppose a police officer has to report all parking violations written in a month, along with all moving violations. AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables The Variance of a Random Variable The variance and standard deviation are measures of spread for a distribution. For a random variable, we will use ππ₯ 2 to describe the variance. The variance is the average of the squared deviations, π₯ β ππ₯ 2 of the variable π₯ from its mean ππ₯ . ππ₯ 2 = π₯π β ππ₯ 2 ππ The standard deviation would be ππ₯ 2 AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables EQ: What is another name for the Expected Value? 7.2 Means and Variances of Random Variables Rules for Variances β’ Rule 1: If π₯ is a random variable and π & π are fixed numbers, then π 2 π+ππ₯ = π 2 ππ₯ 2 β’ Rule 2: If π₯ & π¦ are independent random variables, then π 2 π₯+π¦ = ππ₯ 2 + ππ¦ 2 π 2 π₯βπ¦ = ππ₯ 2 + ππ¦ 2 AP Statistics Part III β Probability: Foundations for Inference Chapter 7: Random Variables 7.2 Means and Variances of Random Variables Homework:
