Using StatisticsExpected Values of Discrete Random VariablesSum and Linear Composite of Random VariablesBernoulli Random VariableThe Binomial Random VariableThe Poisson DistributionContinuous Random VariablesNormal DistributionRandom Variables & Random Variables & Statistical DistributionsStatistical Distributions
After studying this module you should be able to:After studying this module you should be able to:Distinguish between discrete and continuous random variablesExplain how a random variable is characterized by its probability distributionCompute statistics about a random variableCompute statistics about a function of a random variableCompute statistics about the sum or a linear composite of a random variableIdentify which type of distribution a given random variable is most likely to followSolve problems involving standard distributions using formulasLEARNING OBJECTIVESLEARNING OBJECTIVES
Consider the different possible orderings of boy (B) and girl (G) in four sequential births. There are 2*2*2*2=24 = 16possibilities, so the sample space is: BBBBBGBB GBBB GGBB BBBG BGBG GBBG GGBGBBGB BGGB GBGB GGGBBBGG BGGG GBGG GGGGIf girl and boy are each equally likely [P(G) = P(B) = 1/2], and the gender of each child is independent of that of the previous child, then the probability of each of these 16 possibilities is:(1/2)(1/2)(1/2)(1/2) = 1/16.Using Statistics
Now count the number of girls in each set of four sequential births:BBBB(0)BGBB(1)GBBB(1)GGBB(2)BBBG(1)BGBG(2)GBBG(2)GGBG(3)BBGB(1)BGGB(2)GBGB(2)GGGB(3)BBGG(2)BGGG(3)GBGG(3)GGGG(4)Notice that:•each possible outcome is assigned a single numeric value,•all outcomes are assigned a numeric value, and•the value assigned varies over the outcomes.The count of the number of girls is a random variable:A random variable, X, is a function that assigns a single, but variable, value to each element of a sample space.Random Variables
Random Variables (Continued)BBBBBGBBGBBBBBBGBBGBGGBBGBBGBGBGBGGBGBGBBBGGBGGGGBGGGGGBGGBGGGGG01234XXSample SpacePoints on the Real Line
•Therefore, by a random variable we mean a real number X connected with the outcome of a random experiment E.•E.g. if E consists of 2 tosses of a coin, we may consider the random variable which is the number of heads (0, 1 or 2).Outcome HHHTTHTTValue of X2110RANDOM VARIABLE AND PROBABILITY DISTRIBUTION
•A random variable is an uncertain quantity whose value depends on chance.•A random variable has a probability law – a rule that assigns probabilities to the different values of the random variable.•The probability law, the probability assignment is called probability distribution of the random variable.