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5.3 Random Variables 1 Random Variable Discrete Random Variables Continuous Random Variables Normal Distributions as Probability Distributions Random Variables A random variable is a variable whose values are numerical outcomes of a random experiment. That is, we consider all the outcomes in a sample space S and then associate a number with each outcome 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. The probability histogram is a graph showing this probability distribution… Example: Consider tossing a fair coin 3 times. Define X = the number of heads obtained in 3 tosses X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X 2= 3: HHH X Values 0 1 2 3 Probability 1/8 3/8 3/8 1/8 • The figure below shows how to get the probability distribution of X – use a tree diagram! Each outcome has prob=1/16 (HINT: use the “and” rule to show this), and then use the “or” rule to show that P(X=1) = P(TTTH or TTHT or THTT or HTTT) etc…) Discrete Random Variable 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 on a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xi and their probabilities pi: Values of X: P(X) : x1 p1 x2 p2 x3 p3 … … The probabilities pi must satisfy two requirements: 1. Every probability pi is a number between 0 and 1. 2. The sum of the probabilities is 1. To find the probability of any event regarding X, add the probabilities pi of the particular values xi that make up the event. 4 Continuous Random Variable 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 Y takes on its values in an interval of numbers. The probability distribution of Y is described by a density curve. The probability of any event regarding Y is the area under the density curve and above the values of Y that make up the event. 5 Continuous Probability Models Suppose the random expt. is “choose a number at random between 0 and 1”. We cannot assign probabilities to each individual value because there is an infinite interval of possible values, so we model this with a density curve (that is flat) that gives equal weight to all the numbers between 0 and 1. This continuous probability model assigns probabilities as areas under the flat, uniform density curve. The area under the curve and above any range of values is the probability of an outcome in that range. The total area under the curve equals 1. 6 Example: Find the probability of getting a random number that is less than or equal to 0.5 OR greater than 0.8. P(X ≤ 0.5 or X > 0.8) = P(X ≤ 0.5) + P(X > 0.8) = 0.5 + 0.2 = 0.7 Uniform distribution Normal Probability Models There are many situations where the density curve used to assign probabilities to intervals of outcomes is Normal. Normal distributions can be thought of as probability models: Probabilities can be assigned to intervals of outcomes using the Standard Normal probabilities in Table A. We standardize normal data by calculating z-scores so that any Normal curve N(m,s) can be transformed into the standard Normal curve N(0,1). z 7 (x m) s z Normal Probability Models Women’s heights are Normally distributed with mean 64.5 and standard deviation 2.5 in. What is the probability, if we pick one woman at random, that her height will be between 68 and 70 inches P(68 < X < 70)? Because the woman is selected at random, X is a random variable. z (x m) s As before, we calculate the zscores for 68 and 70. For x = 68", z (68 64.5) 1.4 2.5 For x = 70", z (70 64.5) 2.2 2.5 8 5.4 Means and Variances of Random Variables 9 The Mean of a Random Variable The Law of Large Numbers The Variance of a Random Variable HW problems for sections 5.3 and 5.4 The Mean (Expected Value) of a Random Variable The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. This reflects the fact that all outcomes may not be equally likely. The mean is also called the expected value. Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products – so it’s a weighted sum of the r.v.’s values, the weights being the probabilities. 10 μx E ( X ) x1 p1 x2 p2 x3 p3 ... xi pi • We’ve already discussed the mean of a density curve as being the “balance point” of the curve… For a discrete r.v., we’ll compute the mean (or expected value) as a weighted average of the values of X, the weights being the corresponding probabilities and in either case (discrete or continuous), the interpretation of the mean is as the long-run average value of X (in a large number of repetitions of the experiment giving rise to X). • Example: The mean # of Hs in 4 tosses of a fair coin is computed as: (1/16)*0 + (4/16)*1 + (6/16)*2 + (4/16)*3 + (1/16)*4 = (32/16) = 2. • What is your mean “winnings” when you play the “Pick 3” lottery? What does it mean? The Law of Large Numbers Law of Large Numbers: Essentially states that if you sample from a population with mean = m, then the sample mean (xbar) will approximate m for large enough sample sizes. Or that m can be thought of as the expected value (long-run average value) of many independent observations on the variable. Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. The law of large numbers says that as you increase the number of observations, the sample mean of the observed values gets closer and closer to the mean µ of the population. 12 Variance of a 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 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. Variance of a Discrete Random Variable Suppose that X is a discrete random variable whose probability distribution is: Value: x1 x2 x3 … Probability: p1 p2 p3 … and that µX is the mean of X. The variance of X is: Var(X) s X2 (x1 m X ) 2 p1 (x 2 m X ) 2 p2 (x 3 m X ) 2 p3 ... (x i m X ) 2 pi To get the standard deviation of a random variable, take the square root of the variance. 13 HW: Read sections 5.3 and 5.4 (up through p. 273, Law of Large Numbers)… Don’t worry much about the computation of the variance of a discrete random variable… Work on Exercises # 5.43, 5.48-5.55