Stat 200 Exam 2 Study GuideCHAPTER 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 ina population - Family of random variables: consists of all random variables for which the sameformula would be used to find probabilitiesoIn 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) oExample: # of people with Type O blood in a sample of 10 individuals oFor discrete random variables, we can find probabilities for exact outcomes- Probability Notation for a Discrete Random Variable:oX = the random variableoK = a specified number the discrete random variable could assumeoP(X=k) is the probability that X equals k - Probability Distribution of a Discrete Random VariableoProbability Distribution Function (pdf): a table or rule that assigns probabilities to the possible values of the random variable X oConditions 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 ΣkP(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)≤1for any value k oUsing 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 simpleevent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 oExample: Assume the probability of having a girl is ½. 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 tablePDF of X: - Graphing The Probability Distribution FunctionoIt is often useful to represent a probability distribution function with a picture similar to a histogramoThe possible outcome values are placed on the horizontal axis, and their probabilities are placed on the vertical axisoA 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 oA cumulative probability is the probability that the value of a random variable X is less than or equal to a specific valueoThe cumulative distribution function (cdf) for a random variable X is a table or rule that provides the probabilities P(X≤k) for any real number k oFor 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 oExample: 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 P(X≤k)0 1/81 1/8 + 3/8 = 4/82 1/8 + 3/8 + 3/8 = 7/83 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 VariableoIf we know probabilities for all possible values of a random variable, we can determine the mean outcome over the long runoThe 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 variableoExpected Value = Sum of “value x probability”Value is a possible numerical outcome for the random variableProbability is the probability of that outcome Event BBB BBG BGB GBB BGG GBG GGB GGGProb 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8X 0 1 1 1 2 2 2 3K P(=k)0 1/81 3/82 3/83 1/8Compute “value x probability” separately for each possible outcome and then add these quantities to find the expected value oThe notation E(X) represents the mean or expected value of a random variable XThe Greek letter µ can also be usedIn other words, µ = E(X)oIf 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 VariableoThe 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 occuroIt is roughly the average distance the random variable falls from its meanoIf X is a random variable with possible values of x1, x2, x3…, occurring with probabilities p1, p2, p3…, and with expectd value E(X) = µ, thenVariance of X = V(X) = σ2 = (xi - µ)2pi Standard Deviation of X = square root of V(X) = σ = √(xi - µ)2pi- Expected Value and Standard Deviation for a PopulationoSuppose 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 = σ = √(ki - µ)2pi/N = √∑(xi - µ)2pi- Continuous Random Variable: can take any value in an interval or collection of intervals oExample: height for adult women oFor continuous random variables, we cannot find probabilities for exact outcomes. Instead, we are limited to finding probabilities for intervals of values. oThe 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 oIn 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- Notation for Probability in an IntervaloThe two endpoints of an interval are represented by using the letters a and boThe interval of values of X that falls between a and b, including the two endpoints,