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.

3 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 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 VariablesDefinitionA Random variable is a function that maps outcomes of a randomexperiment to real fair coin is tossed 6 times.

5 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,..,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.

6 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 .

7 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 .

8 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 .

9 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?

10 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?