There are 5 cards numbered 1 to 5 , one number on one number on one card . Two cards are drawn at random without replacement . Let X denotes the sum o

Monjur Alahi

There are 5 cards numbered 1 to 5 , one number on one number on one card . Two cards are drawn at random without replacement . Let X denotes the sum of the numbers on two cards drawn . Find the mean and variance of X.


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Salim Ahmed Kawsar

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Here, S={(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(1,4),(4,1),(1,5),(5,1),(2,4),(4,2),(2,5),(5,2),(3,4),(4,3),(3,5),(5,3),(5,4),(4,5)}
`rArrn(S)=20`
Let random variable variable be X which denotes the sum of the numbers on two cards drawn.
X=3,4,5,6,7,8,9
At X=3,P(X)=`2/20=1/10`
At X=4,P(X)`=2/10=1/10`
At X =5,P(X)=`4/20=1/5`
At X=6, P(X)=`4/20=1/5`
At X=7,P(X)=`4/20=1/5`
At X=8, P(X)=`2/20=1/10`
At X=9,P(X)=`2/20=1/10`
`therefore` Mean,E(X)=`SigmaXP(X)=3/10+4/10+5/5+6/5+7/5+8/10+9/10`
=`(3+4+10+12+14+8+9)/10=6`
Also, `SigmaX^(2)P(X)=9/10+16/10+25/5+36/5+49/5+64/10+81/10`
`=(9+16+50+72+98+64+81)/10=39`
`therefore` Var(X)=`SigmaX^(2)P(X)-[SigmaXP(X)]^(2)`
`=39-(6)^(2)=39-36=3`

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