1.

Let X be a discrete random variable whoose probability distribution is defined as follows. P(X=x)={{:(k(x+1)",for x=1,2,3,4"),(2kx",forx=5,6,7"),(0",otherwise"):} where,k is a constant. Calculate (i) the value of k. (ii) E (X). (ii) standard deviation of X.

Answer»

Solution :`P(X=x)={{:(k(x+1)",for x=1,2,3,4"),(2kx",forx=5,6,7"),(0",otherwise"):}`
Thus, we have FOLLOWING table

(i) since, `sumP_(i)=1`
`rArrk(2+3+4+5+10+12+14)=1rArrk=1/50`
(ii) `because E(X)=sumXP(X)`
`therefore E(X)=2k+6k+12k+20k+50k+72k+98k+0=260k`
`=260xx1/50=26/5=5.2` [`becausek=1/50`] ....(i)
(iii) We know that,
Var(X)=`[E(X^(2))]-[E(X)]^(2)=sumX^(2)P(X)-[sum{XP(X)}]^(2)`
`=[2 k+12 k+36k+80K+250k+432k+686k+0]-[5.2]^(2)` [using Eq. (i)]
`=[1498k]-27.04=[1498xx1/50]-27.04`[`becausek=1/50`]
=29.96-27.04=2.92
We know that , STANDARD deviation of X=`SQRT(Var(X))=sqrt(2.92)-1.7088=1.7`(approx)


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