6.1 Discrete and Continuous Random Variables Objectives SWBAT: COMPUTE probabilities using the probability distribution of a discrete random variable. CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. CALCULATE and INTERPRET the standard deviation of a discrete random variable. COMPUTE probabilities using the probability distribution of certain continuous random variables. What is a random variable? Give some examples. A random variable takes numerical values that describe the outcomes of some chance process. X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH Value0123 Probability1/83/8 1/8 Consider tossing a fair coin 3 times. Define X = the number of heads obtained Let X = the score on hole #13 at Augusta National golf course for a randomly selected golfer on day 1 of the 2011 Masters. What is a probability distribution? The probability distribution of a random variable gives its possible values and their probabilities. Probability distributions could be in table or histogram form. Example: Lets define the random variable X as the number of games played in a randomly selected World Series. As a histogram: Its probability distribution: What is a discrete random variable? Give some examples. There are two main types of random variables: discrete and continuous. If we can find a way to list all possible outcomes for a random variable and assign probabilities to each one, we have a discrete random variable. A discrete random variable X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values x i and their probabilities p i : Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 The probabilities p i must satisfy two requirements: 1.Every probability p i is a number between 0 and 1. 2.The sum of the probabilities is 1. To find the probability of any event, add the probabilities p i of the particular values x i that make up the event. Even though it takes a fixed set of values, it could be an infinite set (for example a geometric distribution.more to come on this). To illustrate gaps between, think about shoe size. Shoe size usually goes by halves, i.e. 8, 8.5, 9, 9.5, 10 There are gaps between these values because you cannot get a shoe in size 8.1 or size Compare this to measuring someones foot length. There would be no gaps in measuring foot length because someones foot could measure Foot length is known as a continuous random variable. Think about the gaps between as if you were placing the distribution on a number line, and there would be gaps in between the values on a number line. For example, the variable X in the coin-tossing example is a discrete random variable because there are gaps between the possible values of 0, 1, 2, and 3 on a number line, and their probabilities added to 1. Often, discrete random variables are things you can count. How many languages? Imagine selecting a U.S. high school student at random. Define the random variable X = number of languages spoken by the randomly selected student. The table below gives the probability distribution of X, based on a sample of students from the U.S. Census at School database. a)Show that the probability distribution for X is legitimate. All the probabilities are between 0 and 1 and they add to 1, so this is a legitimate probability distribution. b) Make a histogram of the probability distribution. Describe what you see. Shape: skewed right Center: The median is 1 (more than half the distribution is 1), but the mean will be slightly higher due to the skewness. Spread: The number of languages varies from 1 to 5, but nearly all of the students speak just one or two languages. c) What is the probability that a randomly selected student speaks at least 3 languages? More than 3? Roulette: One wager players can make in Roulette is called a corner bet. To make this bet, a player places his chips on the intersection of four numbered squares on the Roulette table. If one of these numbers comes up on the wheel and the player bet $1, the player gets his $1 back plus $8 more. Otherwise, the casino keeps the original $1 bet. If X = net gain from a single $1 corner bet, the possible outcomes are x = 1 or x = 8. Here is the probability distribution of X: If a player were to make this $1 bet over and over again, what would be the players average gain? In the long run, the player loses $1 in 34 of every 38 games and gains $8 in 4 of every 38 games. Imagine a hypothetical 38 bets. The players average gain is: If a player were to make $1 corner bets many, many times, the average gain would be about $0.05 per bet. In other words, in the long run, the casino keeps about 5 cents of every dollar bet in roulette. How do you calculate the mean (expected value) of a discrete random variable? Is the formula on the formula sheet? The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: This is saying that the mean value of X is equal to the expected value of X, which is equal to the sum of the X values times their probabilities. This is on the formula sheet under probability. Lets go back to the world series example, in which X = the number of games played in a randomly selected World Series. Lets find the expected value of X. How do you interpret the mean (expected value) of a discrete random variable? Lets look at the World Series example. The expected value of X is 5.86 games. How do we interpret this value? If we were to randomly select World Series over and over, the average number of games in the selected Series would be about 5.86. Calculate and interpret the mean of the random variable X in the languages example. If we were to randomly select many, many U.S. high school students at random, the average number of languages spoken would be about Does the expected value of a random variable have to equal one of the possible values of the random variable? Should expected values be rounded? No, the expected value of a random variable does not have to equal one of the possible values of the random variable. Expected values should NOT be rounded. Expected value is really the mean. Think if we were finding the mean of some test scores. It would be perfectly normal to find a mean test score of (usually) would not be one possible value for a test score. We also wouldnt round that mean of Expected value is the same thing as mean, so we wouldnt round. How do you calculate the variance and standard deviation of a discrete random variable? Are these formulas on the formula sheet? Since we use the mean as the measure of center for a discrete random variable, we use the standard deviation as our measure of spread. The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data. Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 and that X is the mean of X. The variance of X is Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 and that X is the mean of X. The variance of X is To get the standard deviation of a random variable, take the square root of the variance. This is on the formula sheet The formula says that you are taking each value of the random variable, subtracting the expected value of the random variable (finding the deviation), squaring that result, and multiplying it by the probability of the random variable. Then, you add up those values. How do you interpret the standard deviation of a discrete random variable? The standard deviation of a random variable X is a measure of how much the values of the variable typically vary from the expected value. In other words, it measures the average distance the outcomes fare from the mean. Skip follow-up question on roulette.go to the languages example. Use your calculator to calculate and interpret the standard deviation of X in the languages example. Reminder: the expected value was Step 2: Enter the values of the random variable in L1 and the corresponding probabilities in L2. Step 1: You must substitute into the formula to show where you calculation is coming from! Step 3: Use one-variable statistics with the values in L1 and the FreqList as L2. The standard deviation is Interpretation: The number of languages spoken by a randomly selected U.S. high school student typically varies by about languages from the mean (1.457). Are there any dangers to be aware of when using the calculator to find the mean and standard deviation of a discrete random variable? You must show some work to get credit. The first couple of terms is fine. What is a continuous random variable? Give some examples. Discrete random variables commonly arise from situations that involve counting something. Situations that involve measuring something often result in a continuous random variable. 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 the event. To think of an example, think about foot length as described before. There are no gaps in between values. Other examples: the amount of time it takes to run the 110 meter hurdles (continuous) vs the number of hurdles cleanly jumped over (discrete) a students age (continuous) and the number of birthdays they have had (discrete) The probability model of a discrete random variable X assigns a probability between 0 and 1 to each possible value of X. A continuous random variable Y has infinitely many possible values. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability. Is it possible to have a shoe size = 8? Is it possible to have a foot length = 8 inches? Yes, it is possible to have a shoe size of 8. No, it is not possible to have a foot length = 8 inches. The reason being is that you can keep measuring down to get more precise, for example inches. How many possible foot lengths are there? How can we graph the distribution of foot length? There are an infinite number of foot lengths. To graph, we can create a histogram with an infinite number of really skinny rectangles that add to 1. This ends up looking like a density curve (think Chapter 2). This is an area where statistics and calculus overlap. You will see when you start examining integrals. How do we find probabilities for continuous random variables? We find probabilities by examining the area under a density curve. Step 2: Step 3: There is about a 57.71% chance that the randomly selected three-year- old female will weigh at least 30 pounds. b) Find the probability that a randomly selected three-year-old female weighs between 25 and 35 pounds. Step 1: N(30.7, 3.6) Step 2: Step 3: There is about a 82.72% chance that the randomly selected three-year- old female will weigh between 25 and 35 pounds. c) If P(X