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Chapter 16: Discrete and Continuous Random Variables Variable o Quantity whose value changes Discrete Variable o Variable whose value is obtained by counting Number of students present Number of heads when flipping three coins Continuous Variable o Variable whose value is obtained by measuring Height of students in class Time it takes to get to school Random Variable o Variable whose value is a numerical outcome of a random phenomenon o Denoted with a capital letter o Probability distribution of a random variable X tells what the possible value of X are and how the probabilities are assigned to those values o Random variable can be discrete or continuous Discrete Random Variable o X has a countable number of possible values o To graph the probability discrete random variable make a frequency table Let X represent the sum of two dice Continuous Random Variable o X takes all values in a given interval of numbers o The probability distribution of a continuous random variable is shown by ****** o Probability that X is between an interval of numbers is the area under the density curve between the interval endpoints Means and Variances of Random Variables o Mean of a discrete variable X is it’s weighted average Each value of x is weighted by its probability o To find the mean of X Multiply each value of X by its probability Then add all the products o The mean of a random variable X is called the ******* of X Law of Large Numbers o As the number of observations increase, the mean of observed values x- bar approaches the mean of the population As sample size increases the sample mean gets closer to the true mean (µ) o The more variation in the outcomes, the more trials are needed to ensure x- bar is close to mew (µ) or the true mean The wider the variation the more sample sizes it’s going to take to get the sample mean to the true mean Rules for Means o If X is a random variable and a and b are fixed numbers 𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑥 o If X and Y are random variables 𝜇𝑋+𝑌 = 𝜇𝑥 + 𝜇𝑌 Variance of a Discrete Random Variable o If X is a discrete random variable with mean (µ) o 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑋 (𝜎𝑋2 = ∑(𝑥𝑖 − 𝜇𝑥 )2 𝑃𝑖 o Standard deviation 𝜎𝑥 is the square root of the variance Rules for Variances o If X is a random variable and a and b are fixed numbers 2 𝜎𝑎+𝑏𝑋 = 𝑏 2 𝜎𝑋2 o If X and Y are independent random variables 2 𝜎𝑋+𝑌 = 𝜎𝑋2 + 𝜎𝑌2 2 𝜎𝑋−𝑌 = 𝜎𝑋2 + 𝜎𝑌2