Stat 200 Exam 2 Study Guide CHAPTER 8 RANDOM VARIABLES Random Variable assigns a number or symbol to each outcome of a random circumstance or equivalently a random variable assigns a number to each unit in a population Family of random variables consists of all random variables for which the same formula would be used to find probabilities o In considering random variables the first step is to identify how the random variables fit into any known family Discrete Random Variables can take one of a countable list of distinct values can only result in a countable set of possibilities o Example of people with Type O blood in a sample of 10 individuals For discrete random variables we can find probabilities for exact outcomes o Probability Notation for a Discrete Random Variable o X the random variable o K a specified number the discrete random variable could assume P X k is the probability that X equals k Probability Distribution of a Discrete Random Variable o o Probability Distribution Function pdf a table or rule that assigns probabilities to the possible values of the random variable X o Conditions for Probabilities for Discrete Random Variables Condition One The sum of the probabilities over all possible values of a discrete random variable must be equal to one or kP X k 1 Condition Two The probability of any specific outcome for a discrete random variable must be between 0 and 1 or 0 P X k 1 for any value k o Using the Sample Space to Find Probabilities for Discrete Random Variables Step 1 list all simple events in the sample space Step 2 Identify the value of the random variable X for each simple event Step 3 Find the probability for each simple event Step 4 To find P X k add the probabilities for all simple events where X k o Example Assume the probability of having a girl is Let X the number of girls in a family with 3 children What is the pdf of X Step 1 possible simple outcomes are 0 1 2 or 3 girls Step 2 X the number of girls in a family with 3 children Step 3 Draw a tree diagram with the probability for each simple event Step 4 Make a table Event Prob X BBB 1 8 0 BBG BGB GBB 1 8 1 8 1 1 1 8 1 BGG GBG GGB GGG 1 8 2 1 8 2 1 8 2 1 8 3 PDF of X K P k 0 1 2 3 1 8 3 8 3 8 1 8 Graphing The Probability Distribution Function o It is often useful to represent a probability distribution function with a picture similar to a histogram o The possible outcome values are placed on the horizontal axis and their probabilities are placed on the vertical axis o A bar is drawn centered on each possible value with the height of the bar equal to the probability for that value The Cumulative Distribution Function of a Discrete Random Variable o A cumulative probability is the probability that the value of a random variable X is less than or equal to a specific value o The cumulative distribution function cdf for a random variable X is a o table or rule that provides the probabilities P X k for any real number k For a discrete random variable the cumulative probability P X k is the sum of all probabilities for all values of X less than or equal to k o Example In the previous example above we found the pdf for X number of girls among three children in a family For each specific value of X the cumulative probability is the sum of the probabilities for all values less than or equal to that value The cdf for X is calculated as follows k 0 1 2 3 P X k 1 8 1 8 3 8 4 8 1 8 3 8 3 8 7 8 1 Note that the cumulative probability for X 3 must equal 1 because all possible values of X are less than or equal to 3 Calculating Expected Value of a Discrete Random Variable o If we know probabilities for all possible values of a random variable we can determine the mean outcome over the long run o The expected value of a random variable X is the mean value of the variable in the sample space or population of possible outcomes Also interpreted as the mean value that would be obtained from an infinite number of observations of the random variable o Expected Value Sum of value x probability Value is a possible numerical outcome for the random variable Probability is the probability of that outcome o o o o o Compute value x probability separately for each possible outcome and then add these quantities to find the expected value o The notation E X represents the mean or expected value of a random variable X The Greek letter can also be used In other words E X If X is a discrete random variable with possible values x1 x2 x3 occurring with probabilities p1 p2 p3 then the expected value of X is calculated as E X xipi Calculating Standard Deviation of a Discrete Random Variable o The standard deviation of a discrete random variable quantifies how spread out the possible values of a discrete random variable might be weighted by how likely each value is to occur It is roughly the average distance the random variable falls from its mean If X is a random variable with possible values of x1 x2 x3 occurring with probabilities p1 p2 p3 and with expectd value E X then Variance of X V X 2 xi 2pi Standard Deviation of X square root of V X xi 2pi Expected Value and Standard Deviation for a Population Suppose a population has N individuals and a measurement X is of interest ki value of X for individual i x1 x2 x3 as the distinct possible values for the measurement X p1 p2 p3 as the proportions of the population with the values x1 x2 x3 Population Mean E X 1 n ki xipi Standard Deviation ki 2pi N xi 2pi Continuous Random Variable can take any value in an interval or collection of intervals o Example height for adult women For continuous random variables we cannot find probabilities for exact outcomes Instead we are limited to finding probabilities for intervals of values o The pdf for a continuous random variable X is a curve such that the area under the curve over an interval equals the probability that X is in that interval In other words the probability that X is between values a and b is the area under the density curve over the interval between the value a and b o Notation for Probability in an Interval o The two endpoints of an interval are represented by using the letters a and b o The interval of values of X that falls between a and b including the two endpoints is written as a X b o The probability …