A discrete random variable X has the probability distribution as given below

(i) Find the value of k.
(ii) Determine the mean of the distribution.

Explanation

We have,

(i) We know that, `sum_(i=1)^(n)P_(i)=1,` where `P_(i)ge0`
`rArrP_(1)+P_(2)+P_(3)+P_(4)=1`
`rArrk+k^(2)+2k^(2)+k=1`
`rArr3k^(2)+2k-1=0`
`rArr3k^(2)+3k-k-1=0`
`rArr (3k-1)(k+1)=0`
`rArrk=1//3rArrk=-1`
Since, k is `ge0rArrk=1//3`
(ii) Mean of the distribution (`mu`)=E(X)=`sum_(i=1_(i))^(n)x_(i)P_(i)`
`=0.5(k)+1(k^(2)+1.5(2k^(2))+2(k)=4k^(2)+2.5k`
`=4cdot1/9+2.5cdot1/3`
`=(4+7.5)/9=23/18`