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Three numbers are selected at random (without replacement) from first six positive integers. Leo, find the mean and variance of the distribution. - Mathematics

Sum

Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X.Also, find the mean and variance of the distribution.

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Solution

X = larger of of three numbers

X = 3, 4, 5, 6

`P(x=3)=6xx1/6xx1/5xx1/4=1/20`

P (x = 4) = `18xx1/6xx1/5xx1/4=3/20`

P (x = 5) = `36xx1/6xx1/5xx1/4=6/20`

P (x = 6) = `60xx1/6xx1/5xx1/4=10/20`

Xi Pi PiXi PiXi2
3

`1/20`

`3/20`

`9/20`

4

`3/20`

`12/20`

`48/20`

5

`6/20`

`30/20`

1

`50/20`

6

`10/20`

`60/20`

`360/20`

`"Mean"=sumP_iX_i=(105/20)=5.25`

`sumP_iX_i^2=567/20`

Var(X)= `sumP_iX_i^2-(sumP_iX_i)^2`

`=567/20-(105/20)^2=0.787`

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