Discrete Random Variables - HAMILTON INSTITUTE

1 Discrete Random VariablesOctober 7, 2010 Discrete Random VariablesRandom VariablesIn many situations, we are interested in numbers associated withthe outcomes of a Random experiment. For example:Testing cars from a production line, we are interested invariables such as average emissions, fuel consumption,acceleration time etcA box of 6 eggs is rejected if it contains one or more brokeneggs. If we examine 10 boxes of eggs, we may be interested in1X1- the number of broken eggs in the 10 boxes2X2- the number of boxes rejectedAn office phone system has 50 lines available and we areinterested in monitoring the number of lines in use at a Random VariablesRandom VariablesIn many situations, we are interested in numbers associated withthe outcomes of a Random experiment.

2 For example:Testing cars from a production line, we are interested invariables such as average emissions, fuel consumption,acceleration time etcA box of 6 eggs is rejected if it contains one or more brokeneggs. If we examine 10 boxes of eggs, we may be interested in1X1- the number of broken eggs in the 10 boxes2X2- the number of boxes rejectedAn office phone system has 50 lines available and we areinterested in monitoring the number of lines in use at a Random VariablesRandom VariablesIn many situations, we are interested in numbers associated withthe outcomes of a Random experiment. For example:Testing cars from a production line, we are interested invariables such as average emissions, fuel consumption,acceleration time etcA box of 6 eggs is rejected if it contains one or more brokeneggs.

3 If we examine 10 boxes of eggs, we may be interested in1X1- the number of broken eggs in the 10 boxes2X2- the number of boxes rejectedAn office phone system has 50 lines available and we areinterested in monitoring the number of lines in use at a Random VariablesRandom VariablesDefinitionA Random variable is a function that maps outcomes of a randomexperiment to real fair coin is tossed 6 times. The number of heads that come up isan example of a Random 3,THHTTT Random Variables can only take values between 0 and 6. Theset of possible values of a Random Variables is known as its Random VariablesRandom VariablesDefinitionA Random variable is a function that maps outcomes of a randomexperiment to real fair coin is tossed 6 times.

4 The number of heads that come up isan example of a Random 3,THHTTT Random Variables can only take values between 0 and 6. Theset of possible values of a Random Variables is known as its Random VariablesRandom VariablesDefinitionA Random variable is a function that maps outcomes of a randomexperiment to real fair coin is tossed 6 times. The number of heads that come up isan example of a Random 3,THHTTT Random Variables can only take values between 0 and 6. Theset of possible values of a Random Variables is known as its Random VariablesRandom VariablesExampleA box of 6 eggs is rejected once it contains one or more brokeneggs. If we examine 10 boxes of eggs and define the randomvariablesX1,X2as1X1- the number of broken eggs in the 10 boxes2X2- the number of boxes rejectedThen the range ofX1is{0,1,2.}

5 ,59,60}, while the range ofX2is 0,1,2,..,10}. Discrete Random VariablesRandom VariablesExampleA box of 6 eggs is rejected once it contains one or more brokeneggs. If we examine 10 boxes of eggs and define the randomvariablesX1,X2as1X1- the number of broken eggs in the 10 boxes2X2- the number of boxes rejectedThen the range ofX1is{0,1,2,..,59,60}, while the range ofX2is 0,1,2,..,10}. Discrete Random VariablesContinuous and Discrete Random VariablesIf the range of a Random variable is finite or countably infinite,it is said to be a Discrete Random variable . For example -Number of broken eggs in a batch or the number of bits inerror in a transmitted the range of a Random variable is continuous, it is said to bea continuous Random variable .

6 For example - the current in acopper wire or the length of a manufactured Variables are usually denoted by capital lettersX. Thevalues of the Variables are usually denoted by lower case notationP(X=x)stands for the probability that the Random variableXtakes Random VariablesContinuous and Discrete Random VariablesIf the range of a Random variable is finite or countably infinite,it is said to be a Discrete Random variable . For example -Number of broken eggs in a batch or the number of bits inerror in a transmitted the range of a Random variable is continuous, it is said to bea continuous Random variable . For example - the current in acopper wire or the length of a manufactured Variables are usually denoted by capital lettersX.

7 Thevalues of the Variables are usually denoted by lower case notationP(X=x)stands for the probability that the Random variableXtakes Random VariablesContinuous and Discrete Random VariablesIf the range of a Random variable is finite or countably infinite,it is said to be a Discrete Random variable . For example -Number of broken eggs in a batch or the number of bits inerror in a transmitted the range of a Random variable is continuous, it is said to bea continuous Random variable . For example - the current in acopper wire or the length of a manufactured Variables are usually denoted by capital lettersX. Thevalues of the Variables are usually denoted by lower case notationP(X=x)stands for the probability that the Random variableXtakes Random VariablesProbability Mass FunctionsDefinitionFor a Discrete Random variableX, its probability mass functionf( )is specified by giving the valuesf(x) =P(X=x) for allxin therange is the probability mass function of the Random variable thatcounts the number of heads on 3 tosses of a fair coin?

8 The range of the variable is{0,1,2,3}.P(X= 0) = (12)3P(X= 1) = 3(12)3P(X= 2) = 3(12)3P(X= 3) = (12)3 Discrete Random VariablesProbability Mass FunctionsDefinitionFor a Discrete Random variableX, its probability mass functionf( )is specified by giving the valuesf(x) =P(X=x) for allxin therange is the probability mass function of the Random variable thatcounts the number of heads on 3 tosses of a fair coin?The range of the variable is{0,1,2,3}.P(X= 0) = (12)3P(X= 1) = 3(12)3P(X= 2) = 3(12)3P(X= 3) = (12)3 Discrete Random VariablesProbability Mass FunctionsDefinitionFor a Discrete Random variableX, its probability mass functionf( )is specified by giving the valuesf(x) =P(X=x) for allxin therange is the probability mass function of the Random variable thatcounts the number of heads on 3 tosses of a fair coin?

9 The range of the variable is{0,1,2,3}.P(X= 0) = (12)3P(X= 1) = 3(12)3P(X= 2) = 3(12)3P(X= 3) = (12)3 Discrete Random VariablesProbability Mass FunctionsDefinitionFor a Discrete Random variableX, its probability mass functionf( )is specified by giving the valuesf(x) =P(X=x) for allxin therange is the probability mass function of the Random variable thatcounts the number of heads on 3 tosses of a fair coin?The range of the variable is{0,1,2,3}.P(X= 0) = (12)3P(X= 1) = 3(12)3P(X= 2) = 3(12)3P(X= 3) = (12)3 Discrete Random VariablesProbability Mass FunctionsExampleConsider the following game. A fair 4-sided die, with the numbers1,2,3,4 is rolled twice. If the score on the second roll is strictlygreater than the score on the first the player wins the difference ineuro.

10 If the score on the second roll is strictly less than the scoreon the first roll, the player loses the difference in euro. If the scoresare equal, the player neither wins nor loses. If we letXdenote the(possibly negative) winnings of the player, what is the probabilitymass function ofX? (Xcan take any of the values 3, 2, 1,0,1,2,3.) Discrete Random VariablesExampleExampleThe total number of outcomes of the experiment is 4 4 = (X= 0):Xwill take the value 0 for the outcomes(1,1),(2,2),(3,3),(4,4). Sof(0) = (X= 1):Xwill take the value 1 for the outcomes(1,2),(2,3),(3,4). Sof(1) = (X= 2):Xwill take the value 2 for the outcomes(1,3),(2,4). Sof(2) = (X= 3): Similarlyf(3) = we find the probability mass function is Continuing inthe same way we see that the probability mass function isx-3-2-10123f(x)116216316416316216116 Discrete Random VariablesProbability Mass FunctionsA functionfcan only be a probability mass function if it satisfiescertain conditions.