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Random Variables A random variable assumes a value based on the outcome of a random event. ◦ We use a capital letter, like X, to denote a random variable. ◦ A particular value of a random variable will be denoted with a lower case letter, in this case x. There are two types of random variables: ◦ Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours ◦ Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term Example A professor asked his introductory statistics students to state how many siblings they have. The result is organized in the following table. Let X denote the number of siblings of a randomly selected student. a. Determine the probability distribution of the random variable X. b. Construct a probability histogram for the random variable X. Solution Example The table displays these probabilities and provides the probability distribution. Example Toss a balanced dime three times and record the number of heads. Let X be the number of heads of each three-toss experiment. a. Repeat the experiment 1000 times and record the frequencies and proportions for the number of heads. b. Determine the probability distribution of the random variable X. c. Construct a probability histogram for the random variable X. Solution Example a) We used a computer to simulate 1000 observations of the random variable X, the number of heads obtained in three tosses of a balanced dime. The following table shows the frequencies and proportions for the numbers of heads obtained in the 1000 observations. Solution Example c) This result is more easily seen if we compare the proportion histogram to the probability histogram of the random variable X. A probability model for a random variable consists of: ◦ The collection of all possible values of a random variable, and ◦ the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value. The expected value of a (discrete) random variable can be found by summing the products of each possible value and the probability that it occurs: E X x P x An American Roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. The dealer spins the wheel and whirls a small ball in the opposite direction within the wheel. Gamblers bet on where the ball will come to rest. One of the simplest bet to choose red (or black). A bet of $1 on red pays off an additional $1 if the ball lands in a red slot. Otherwise, the player loses his $1. Is it a fair game? Insurance company make bets. They bet that you’re going to live a long life. You bet that you’re going to die sooner. How to find a “fair price” for this kind of bet? Suppose an insurance company offers a “death and disability” policy that pays $10,000 when you die or $5000 if you are permanently disabled. How can the insurance company determine the annual premium? Suppose the insurance company insures 1000 people. The death rate and disability rate in any year can be assumed as follows: Policyholder outcome Probability P (X=x) Payout Death 1/1000 10,000 Disability 2/1000 5000 977/1000 0 Neither Let X be the insurance company’s payout in a given year. For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with discrete random variables as well. The variance for a random variable is: 2 Var X x P x 2 The standard deviation for a random variable is: SD X Var X Adding or subtracting a constant from data shifts the mean but doesn’t change the variance or S.D: E(X ± c) = E(X) ± c Var(X ± c) = Var(X) ◦ Example: Consider everyone in a company receiving a $5000 increase in salary. In general, multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant: E(aX) = aE(X) Var(aX) = a2Var(X) ◦ Example: Consider everyone in a company receiving a 10% increase in salary. The mean of the sum (or difference) of two random variables is the sum (or difference) of the means. E(X ± Y) = E(X) ± E(Y) If the random variables are independent, the variance of their sum or difference is always the sum of the variances. Var(X ± Y) = Var(X) + Var(Y) Combining Random Variables (The Bad News) It would be nice if we could go directly from models of each random variable to a model for their sum. But, the probability model for the sum of two random variables is not necessarily the same as the model we started with even when the variables are independent. Thus, even though expected values may add, the probability model itself is different. Combining Random Variables (The good news) Nearly everything we’ve said about how discrete random variables behave is true of continuous random variables, as well. When two independent continuous random variables have Normal models, so does their sum or difference. This fact will let us apply our knowledge of Normal probabilities to questions about the sum or difference of independent random variables. Consider the company that manufactures and ships small stereo systems. The times required to pack the stereos can be described by a Normal model with a mean of 9 minutes and standard deviation of 1.5 minutes. The times for the boxing stage can also be modeled as Normal, with a mean of 6 minutes and standard deviation of 1 minute. Find the expected total times for packing and boxing a system. Find the standard deviation of packing and boxing time . Find the expected time for packing two systems ◦ What’s the probability that packing two systems takes over 20 minutes? ◦ What percentage of stereo system take longer to pack than to box? Page 427 – 430 Problem # 1–15 odd, 19, 23, 25, 27, 29, 35, 43.