+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 6: Random Variables Section 6.1 Discrete and Continuous Random Variables

+ Sample Spaces In Chapter 5, we studied sample spaces. If we toss two coins, the sample space is {HH, HT, TH, or TT}. We statisticians like numbers. So, let’s convert this sample space to numbers by counting the number of heads. Then the sample space is {0, 1, 2}.

+ Discrete and Continuous Random Variables Random Variable and Probability Distribution A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur. A numerical variable that describes the outcomes of a chance process is called a random variable. The probability model for a random variable is its probability distribution Definition: A random variable takes numerical values that describe the outcomes of some chance process. The probability distribution of a random variable gives its possible values and their probabilities. Example: Consider tossing a fair coin 3 times. Define X = the number of heads obtained X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH Value0123 Probability1/83/8 1/8

+ Discrete Random Variables 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, wehave a discrete random variable. Discrete and Continuous Random Variables 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. Discrete Random Variables and Their Probability Distributions

+ Example: Babies’ Health at Birth Read the example on page 343.(a) Show that the probability distribution for X is legitimate. (b) Make a histogram of the probability distribution. Describe what you see. (c) Apgar scores of 7 or higher indicate a healthy baby. What is P ( X ≥ 7)? (a) All probabilities are between 0 and 1 and they add up to 1. This is a legitimate probability distribution. (b) The left-skewed shape of the distribution suggests a randomly selected newborn will have an Apgar score at the high end of the scale. There is a small chance of getting a baby with a score of 5 or lower. (c) P(X ≥ 7) =.908 We’d have a 91 % chance of randomly choosing a healthy baby. Value: Probability:

+ Mean of a Discrete Random Variable When analyzing discrete random variables, we’ll follow the same strategy we used with quantitative data – describe the shape,center, and spread, and identify any outliers. The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by itsprobability. Discrete and Continuous Random Variables Definition: 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:

+ Example: Apgar Scores – What’s Typical? Consider the random variable X = Apgar Score Compute the mean of the random variable X and interpret it in context. Value: Probability: The mean Apgar score of a randomly selected newborn is This is the long- term average Agar score of many, many randomly chosen babies. Note: The expected value does not need to be a possible value of X or an integer! It is a long-term average over many repetitions.

+ Toss 4 Coins Let X = the number of Heads in 4 coin tosses. Write the probability distribution of X. Graph the probability distribution. Describe the graph. Find the probability of tossing at least two heads. Find the probability of tossing at least one head. Find the expected number of heads.

+ Standard Deviation of a Discrete Random Variable Since we use the mean as the measure of center for a discrete random variable, we’ll use the standard deviation as our measure ofspread. The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data. Discrete and Continuous Random Variables Definition: 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 µ 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.

+ Example: Apgar Scores – How Variable Are They? Consider the random variable X = Apgar Score Compute the standard deviation of the random variable X and interpret it in context. Value: Probability: The standard deviation of X is On average, a randomly selected baby’s Apgar score will differ from the mean by about 1.4 units. Variance

+ Calculator Investigation Let’s choose 200 random integers between 1 and 10 and store them in L1. MATH/PRB/5:RandInt(1,10,200)STO->L1. STAT/2: SortA(L1) Scroll through your list. Do any numbers repeat? What’s the probability of choosing a 1? What’s the probability of choosing a number between 3 and 5?

+ Calculator Investigation 2 Let’s choose 200 random numbers and store them in L2. MATH/PRB/1:Rand(200)STO->L2. STAT/2: SortA(L2) Scroll through your list. Do any numbers repeat? What’s the probability of choosing ?

+ Discrete and Continuous Random Variables There are two types of random variables: discrete and continuous. Discrete random variables have a finite (countable) number of possible values. They usually arise out of COUNTING something. The table of possible values of x and the associated probabilities is called a PROBABILITY DISTRIBUTION. We use a histogram to graph a discrete random variable. Continuous random variables take on all values in an interval of numbers. So, there are an infinite number of possible values. They usually arise out of MEASURING something. The graph of a continuous RV is a density curve.

+ The weird thing about continuous RVs When you instructed the calculator to pick 200 random numbers, what is the probability that was in your list? That’s because all continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability. That’s why on a normal curve, there is no difference between the answer P(X<85) and P(X≤85). The probability that X = 85 is zero, so it doesn’t change the answer.

+ Example: Young Women’s Heights Read the example on page 351. Define Y as the height of a randomly chosen young woman. Y is a continuous random variable whose probability distribution is N (64, 2.7). What is the probability that a randomly chosen young woman has height between 68 and 70 inches? P(68 ≤ Y ≤ 70) = ??? P(1.48 ≤ Z ≤ 2.22) = P(Z ≤ 2.22) – P(Z ≤ 1.48) = – = There is about a 5.6% chance that a randomly chosen young woman has a height between 68 and 70 inches.

+ Normal Curve Review Suppose a study investigating the effects of car speed on accident severity reveal that the speed in fatal automobile accidents was distributed normally with a mean of 45 mph with a standard deviation of 15 mph. Sketch a normal curve for this situation.

+ Suppose a study investigating the effects of car speed on accident severity reveal that the speed in fatal automobile accidents was distributed normally with a mean of 45 mph with a standard deviation of 15 mph. Complete the sentence: approximately 95% of speeds fall between _____ and ______. What proportion of accidents involve vehicle speeds over 65 mph?

+ Suppose a study investigating the effects of car speed on accident severity reveal that the speed in fatal automobile accidents was distributed normally with a mean of 45 mph with a standard deviation of 15 mph. What proportion of accidents involve vehicle speeds between 35 and 45 mph? What vehicle speed marks the top 7% of all vehicle speeds?